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Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan

Chapter 11

Sequences; Induction; the Binomial Theorem - all with Video Answers

Educators

Chapter Questions

00:41

Problem 1

For the function $f(x)=\frac{x-1}{x}$, find $f(2)$ and $f(3)$.

Julie Silva
Julie Silva
Numerade Educator
00:00

Problem 1

In $\mathrm{a}(\mathrm{n})$ ___________ sequence, the difference between successive terms is a constant.

Rebecca Dias
Rebecca Dias
Numerade Educator
01:29

Problem 1

If $$\$ 1000$$ is invested at $4 \%$ per annum compounded semiannually, how much is in the account after 2 years?

Dale Sanford
Dale Sanford
Numerade Educator
03:01

Problem 1

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
2+4+6+\cdots+2 n=n(n+1)
$$

Gregory Higby
Gregory Higby
Numerade Educator
01:08

Problem 1

The ________ _________ is a triangular display of the binomial coefficients.

John Vanschoick
John Vanschoick
Numerade Educator
00:33

Problem 2

True or False A function is a relation between two sets $D$ and $R$ so that each element $x$ in the first set $D$ is related to exactly one element $y$ in the second set $R$.

Julie Silva
Julie Silva
Numerade Educator
00:32

Problem 2

True or False For an arithmetic sequence $\left\{a_n\right\}$ whose first term is $a_1$ and whose common difference is $d$, the $n$th term is determined by the formula $a_n=a_1+n d$.

Julie Silva
Julie Silva
Numerade Educator
01:56

Problem 2

How much do you need to invest now at $5 \%$ per annum compounded monthly so that in 1 year you will have $$\$ 10,000$$ ?

Dale Sanford
Dale Sanford
Numerade Educator
03:06

Problem 2

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+5+9+\cdots+(4 n-3)=n(2 n-1)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:43

Problem 2

$$
\left(\begin{array}{l}
n \\
0
\end{array}\right)=\_\_ \text { and }\left(\begin{array}{l}
n \\
1
\end{array}\right)=\_\_ \text {. }
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:17

Problem 3

$\mathrm{A}(\mathrm{n})$ _______ is a function whose domain is the set of positive integers.

Julie Silva
Julie Silva
Numerade Educator
00:32

Problem 3

If the 5th term of an arithmetic sequence is 12 and the common difference is 5 , then the 6 th term of the sequence is ________.

Julie Silva
Julie Silva
Numerade Educator
01:01

Problem 3

In $a(n)$ _________ sequence, the ratio of successive terms is a constant. $\infty$

Amit Srivastava
Amit Srivastava
Numerade Educator
03:45

Problem 3

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
3+4+5+\cdots+(n+2)=\frac{1}{2} n(n+5)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:12

Problem 3

True or False $\left(\begin{array}{l}n \\ j\end{array}\right)=\frac{j !}{(n-j) ! n !}$

Grant Mansfield
Grant Mansfield
Numerade Educator
00:15

Problem 4

True or False The notation $a_5$ represents the fifth term of a sequence.

Julie Silva
Julie Silva
Numerade Educator
00:25

Problem 4

True or False The sum $S_n$ of the first $n$ terms of an arithmetic sequence $\left\{a_n\right\}$ whose first term is $a_1$ can be found using the formula $S_n=\frac{n}{2}\left(a_1+a_n\right)$.

Julie Silva
Julie Silva
Numerade Educator
00:55

Problem 4

If $|r|<1$, the sum of the geometric series $\sum_{k=1}^{\infty} a r^{k-1}$ is _________.
(a) $a(1-r)$
(b) $(1-r)^a$
(c) $a^{1-r}$
(d) $\frac{a}{1-r}$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:07

Problem 4

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
3+5+7+\cdots+(2 n+1)=n(n+2)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:45

Problem 4

The _______ ________ can be used to expand expressions like $\overline{(2 x+3)}$.

John Vanschoick
John Vanschoick
Numerade Educator
00:20

Problem 5

True or False If $n \geq 2$ is an integer, then
$$
n !=n(n-1) \cdots 3 \cdot 2 \cdot 1
$$

Julie Silva
Julie Silva
Numerade Educator
00:19

Problem 5

An arithmetic sequence can always be expressed as a(n) ______ sequence.
(a) Fibonacci
(b) alternating
(c) geometric
(d) recursive

Rebecca Dias
Rebecca Dias
Numerade Educator
00:14

Problem 5

If a series does not converge, it is called a(n) _______ ________.

Dale Sanford
Dale Sanford
Numerade Educator
02:54

Problem 5

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
2+5+8+\cdots+(3 n-1)=\frac{1}{2} n(3 n+1)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:34

Problem 5

Evaluate each expression.
$$
\left(\begin{array}{l}
5 \\
3
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:35

Problem 6

The sequence $a_1=5, a_n=3 a_{n-1}$ is an example of a(n) ________ sequence.
(a) alternating
(b) recursive
(c) Fibonacci
(d) summation

Julie Silva
Julie Silva
Numerade Educator
00:50

Problem 6

If $a_n=-2 n+7$ is the $n$th term of an arithmetic sequence, the first term is _________.
(a) -2
(b) 0
(c) 5
(d) 7

Rebecca Dias
Rebecca Dias
Numerade Educator
00:39

Problem 6

True or False A geometric sequence may be defined recursively.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:36

Problem 6

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+4+7+\cdots+(3 n-2)=\frac{1}{2} n(3 n-1)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:33

Problem 6

Evaluate each expression.
$$
\left(\begin{array}{l}
7 \\
3
\end{array}\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:26

Problem 7

The notation $a_1+a_2+a_3+\cdots+a_n=\sum_{k=1}^n a_k$ is an example of ______ notation.

Julie Silva
Julie Silva
Numerade Educator
01:17

Problem 7

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{s_n\right\}=\{n+4\}
$$

Julie Silva
Julie Silva
Numerade Educator
00:48

Problem 7

True or False In a geometric sequence, the common ratio is always a positive number.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:04

Problem 7

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+2+2^2+\cdots+2^{n-1}=2^n-1
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:36

Problem 7

Evaluate each expression.
$$
\left(\begin{array}{l}
7 \\
5
\end{array}\right)
$$

Julie Silva
Julie Silva
Numerade Educator
00:26

Problem 8

$$\sum_{k=1}^n k=1+2+3+\cdots+n=$$ _______.
(a) $n !$
(b) $\frac{n(n+1)}{2}$
(c) $n k$
(d) $\frac{n(n+1)(2 n+1)}{6}$

Julie Silva
Julie Silva
Numerade Educator
01:13

Problem 8

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{s_n\right\}=\{n-5\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:06

Problem 8

True or False For a geometric sequence with first term $a_1$ and common ratio $r$, where $r \neq 0, r \neq 1$, the sum of the first $n$ terms is $S_n=a_1 \cdot \frac{1-r^n}{1-r}$.

Amit Srivastava
Amit Srivastava
Numerade Educator
02:34

Problem 8

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+3+3^2+\cdots+3^{n-1}=\frac{1}{2}\left(3^n-1\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:17

Problem 8

Evaluate each expression.
$$
\left(\begin{array}{l}
9 \\
7
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:29

Problem 9

Evaluate each factorial expression.
$$
10 !
$$

Julie Silva
Julie Silva
Numerade Educator
01:19

Problem 9

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{a_n\right\}=\{2 n-5\}
$$

Julie Silva
Julie Silva
Numerade Educator
00:53

Problem 9

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{s_n\right\}=\left\{3^n\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:35

Problem 9

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+4+4^2+\cdots+4^{n-1}=\frac{1}{3}\left(4^n-1\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:02

Problem 9

Evaluate each expression.
$$
\left(\begin{array}{l}
50 \\
49
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:24

Problem 10

Evaluate each factorial expression.
$$
9 !
$$

Julie Silva
Julie Silva
Numerade Educator
01:15

Problem 10

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{b_n\right\}=\{3 n+1\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:13

Problem 10

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{s_n\right\}=\left\{(-5)^n\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:22

Problem 10

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1+5+5^2+\cdots+5^{n-1}=\frac{1}{4}\left(5^n-1\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:20

Problem 10

Evaluate each expression.
$$
\left(\begin{array}{c}
100 \\
98
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:49

Problem 11

Evaluate each factorial expression.
$$
\frac{9 !}{6 !}
$$

Julie Silva
Julie Silva
Numerade Educator
01:22

Problem 11

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{c_n\right\}=\{6-2 n\}
$$

Julie Silva
Julie Silva
Numerade Educator
02:50

Problem 11

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{a_n\right\}=\left\{-3\left(\frac{1}{2}\right)^n\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:45

Problem 11

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:39

Problem 11

Evaluate each expression.
$$
\left(\begin{array}{l}
1000 \\
1000
\end{array}\right)
$$

Grant Mansfield
Grant Mansfield
Numerade Educator
00:48

Problem 12

Evaluate each factorial expression.
$$
\frac{12 !}{10 !}
$$

Julie Silva
Julie Silva
Numerade Educator
01:30

Problem 12

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{a_n\right\}=\{4-2 n\}
$$

Julie Silva
Julie Silva
Numerade Educator
02:28

Problem 12

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{b_n\right\}=\left\{\left(\frac{5}{2}\right)^n\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:02

Problem 12

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:03

Problem 12

Evaluate each expression.
$$
\left(\begin{array}{c}
1000 \\
0
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:52

Problem 13

Evaluate each factorial expression.
$$
\frac{3 ! 7 !}{4 !}
$$

Julie Silva
Julie Silva
Numerade Educator
03:19

Problem 13

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{t_n\right\}=\left\{\frac{1}{2}-\frac{1}{3} n\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
02:25

Problem 13

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{c_n\right\}=\left\{\frac{2^{n-1}}{4}\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:08

Problem 13

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1^2+2^2+3^2+\cdots+n^2=\frac{1}{6} n(n+1)(2 n+1)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:59

Problem 13

Evaluate each expression.
$$
\left(\begin{array}{l}
55 \\
23
\end{array}\right)
$$

John Vanschoick
John Vanschoick
Numerade Educator
01:28

Problem 14

Solve each linear programming problem.
$$
\frac{5 ! 8 !}{3 !}
$$

James Kiss
James Kiss
Numerade Educator
02:48

Problem 14

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{t_n\right\}=\left\{\frac{2}{3}+\frac{n}{4}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:33

Problem 14

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{d_n\right\}=\left\{\frac{3^n}{9}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:30

Problem 14

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1^3+2^3+3^3+\cdots+n^3=\frac{1}{4} n^2(n+1)^2
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:25

Problem 14

Evaluate each expression.
$$
\left(\begin{array}{l}
60 \\
20
\end{array}\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:28

Problem 15

Write down the first five terms of each sequence.
$$
\left\{s_n\right\}=\{n\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:58

Problem 15

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{s_n\right\}=\left\{\ln 3^n\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:36

Problem 15

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{e_n\right\}=\left\{2^{n / 3}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:55

Problem 15

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
4+3+2+\cdots+(5-n)=\frac{1}{2} n(9-n)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:27

Problem 15

Evaluate each expression.
$$
\left(\begin{array}{l}
47 \\
25
\end{array}\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:54

Problem 16

Write down the first five terms of each sequence.
$$
\left\{s_n\right\}=\left\{n^2+1\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:34

Problem 16

Show that each sequence is arithmetic. Find the common difference, and write out the first four terms.
$$
\left\{s_n\right\}=\left\{e^{\ln n}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:02

Problem 16

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{f_n\right\}=\left\{3^{2 n}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:30

Problem 16

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
-2-3-4-\cdots-(n+1)=-\frac{1}{2} n(n+3)
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:27

Problem 16

Evaluate each expression.
$$
\left(\begin{array}{l}
37 \\
19
\end{array}\right)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:07

Problem 17

Write down the first five terms of each sequence.
$$
\left\{a_n\right\}=\left\{\frac{n}{n+2}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:09

Problem 17

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=2 ; \quad d=3
$$

Julie Silva
Julie Silva
Numerade Educator
02:02

Problem 17

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{t_n\right\}=\left\{\frac{3^{n-1}}{2^n}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:45

Problem 17

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:13

Problem 17

Expand each expression using the Binomial Theorem.
$$
(x+1)^5
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:32

Problem 18

Write down the first five terms of each sequence.
$$
\left\{b_n\right\}=\left\{\frac{2 n+1}{2 n}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:17

Problem 18

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=-2 ; \quad d=4
$$

Julie Silva
Julie Silva
Numerade Educator
01:40

Problem 18

Show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$$
\left\{u_n\right\}=\left\{\frac{2^n}{3^{n-1}}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:04

Problem 18

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
1 \cdot 2+3 \cdot 4+5 \cdot 6+\cdots+(2 n-1)(2 n)=\frac{1}{3} n(n+1)(4 n-1)
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:21

Problem 18

Expand each expression using the Binomial Theorem.
$$
(x-1)^5
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:59

Problem 19

Write down the first five terms of each sequence.
$$
\left\{c_n\right\}=\left\{(-1)^{n+1} n^2\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:08

Problem 19

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=5 ; \quad d=-3
$$

Vishal Parmar
Vishal Parmar
Numerade Educator
01:08

Problem 19

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=2 ; \quad r=3
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:08

Problem 19

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
n^2+n \text { is divisible by } 2 \text {. }
$$

Amy Jiang
Amy Jiang
Numerade Educator
02:02

Problem 19

Expand each expression using the Binomial Theorem.
$$
(x-2)^6
$$

Amy Jiang
Amy Jiang
Numerade Educator
02:50

Problem 20

Write down the first five terms of each sequence.
$$
\left\{d_n\right\}=\left\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:26

Problem 20

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=6 ; \quad d=-2
$$

Julie Silva
Julie Silva
Numerade Educator
01:29

Problem 20

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=-2 ; \quad r=4
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:02

Problem 20

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
n^3+2 n \text { is divisible by } 3 \text {. }
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:29

Problem 20

Expand each expression using the Binomial Theorem.
$$
(x+3)^5
$$

Amy Jiang
Amy Jiang
Numerade Educator
02:03

Problem 21

Write down the first five terms of each sequence.
$$
\left\{s_n\right\}=\left\{\frac{2^n}{3^n+1}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:25

Problem 21

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=0 ; \quad d=\frac{1}{2}
$$

Julie Silva
Julie Silva
Numerade Educator
01:38

Problem 21

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=5 ; \quad r=-1
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:28

Problem 21

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
n^2-n+2 \text { is divisible by } 2 \text {. }
$$

Sirat Shah
Sirat Shah
Numerade Educator
04:05

Problem 21

Expand each expression using the Binomial Theorem.
$$
(3 x+1)^4
$$

John Vanschoick
John Vanschoick
Numerade Educator
01:26

Problem 22

Write down the first five terms of each sequence.
$$
\left\{s_n\right\}=\left\{\left(\frac{4}{3}\right)^n\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:42

Problem 22

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=1 ; \quad d=-\frac{1}{3}
$$

Julie Silva
Julie Silva
Numerade Educator
01:23

Problem 22

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=6 ; \quad r=-2
$$

Dale Sanford
Dale Sanford
Numerade Educator
04:21

Problem 22

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$$
n(n+1)(n+2) \text { is divisible by } 6 \text {. }
$$

Amy Jiang
Amy Jiang
Numerade Educator
04:46

Problem 22

Expand each expression using the Binomial Theorem.
$$
(2 x+3)^5
$$

John Vanschoick
John Vanschoick
Numerade Educator
02:14

Problem 23

Write down the first five terms of each sequence.
$$
\left\{t_n\right\}=\left\{\frac{(-1)^n}{(n+1)(n+2)}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:36

Problem 23

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=\sqrt{2} ; \quad d=\sqrt{2}
$$

Julie Silva
Julie Silva
Numerade Educator
00:34

Problem 23

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=0 ; \quad r=\frac{1}{2}
$$

Sherin Hussain
Sherin Hussain
Numerade Educator
04:49

Problem 23

Prove each statement.
If $x>1$, then $x^n>1$.

Sirat Shah
Sirat Shah
Numerade Educator
01:21

Problem 23

Expand each expression using the Binomial Theorem.
$$
\left(x^2+y^2\right)^5
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:22

Problem 24

Write down the first five terms of each sequence.
$$
\left\{a_n\right\}=\left\{\frac{3^n}{n}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:22

Problem 24

Find the nth term of the arithmetic sequence $\left\{a_n\right\}$ whose initial term a and common difference d are given. What is the 51st term?
$$
a_1=0 ; \quad d=\pi
$$

Julie Silva
Julie Silva
Numerade Educator
01:47

Problem 24

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=1 ; \quad r=-\frac{1}{3}
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:44

Problem 24

Prove each statement.
If $0<x<1$, then $0<x^n<1$.

Sirat Shah
Sirat Shah
Numerade Educator
05:35

Problem 24

Expand each expression using the Binomial Theorem.
$$
\left(x^2-y^2\right)^6
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:45

Problem 25

Write down the first five terms of each sequence.
$$
\left\{b_n\right\}=\left\{\frac{n}{e^n}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:37

Problem 25

Find the indicated term in each arithmetic sequence.
$$
100 \text { th term of } 2,4,6, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
02:09

Problem 25

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=\sqrt{2} ; \quad r=\sqrt{2}
$$

Dale Sanford
Dale Sanford
Numerade Educator
05:49

Problem 25

Prove each statement.
$a-b$ is a factor of $a^n-b^n$.

Sirat Shah
Sirat Shah
Numerade Educator
07:25

Problem 25

Expand each expression using the Binomial Theorem.
$$
(\sqrt{x}+\sqrt{2})^6
$$

John Vanschoick
John Vanschoick
Numerade Educator
01:30

Problem 26

Write down the first five terms of each sequence.
$$
\left\{c_n\right\}=\left\{\frac{n^2}{2^n}\right\}
$$

Julie Silva
Julie Silva
Numerade Educator
01:29

Problem 26

Find the indicated term in each arithmetic sequence.
$$
80 \text { th term of }-1,1,3, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:13

Problem 26

Find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$$
a_1=0 ; \quad r=\frac{1}{\pi}
$$

Dale Sanford
Dale Sanford
Numerade Educator
05:49

Problem 26

Prove each statement.
$a+b$ is a factor of $a^{2 n+1}+b^{2 n+1}$.

Sirat Shah
Sirat Shah
Numerade Educator
07:06

Problem 26

Expand each expression using the Binomial Theorem.
$$
(\sqrt{x}-\sqrt{3})^4
$$

John Vanschoick
John Vanschoick
Numerade Educator
00:48

Problem 27

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:29

Problem 27

Find the indicated term in each arithmetic sequence.
$$
90 \text { th term of } 1,-2,-5, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
00:44

Problem 27

Find the indicated term of each geometric sequence.
$$
7 \text { th term of } 1, \frac{1}{2}, \frac{1}{4}, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
05:46

Problem 27

Prove each statement.
$(1+a)^n \geq 1+n a$, for $a>0$

Sirat Shah
Sirat Shah
Numerade Educator
02:03

Problem 27

Expand each expression using the Binomial Theorem.
$$
(a x+b y)^5
$$

Thomas Emment
Thomas Emment
Numerade Educator
01:04

Problem 28

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:26

Problem 28

Find the indicated term in each arithmetic sequence.
$$
80 \text { th term of } 5,0,-5, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
00:43

Problem 28

Find the indicated term of each geometric sequence.
$$
8 \text { th term of } 1,3,9, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:06

Problem 28

Show that the statement " $n^2-n+41$ is a prime number" is true for $n=1$ but is not true for $n=41$.

Sirat Shah
Sirat Shah
Numerade Educator
05:02

Problem 28

Expand each expression using the Binomial Theorem.
$$
(a x-b y)^4
$$

John Vanschoick
John Vanschoick
Numerade Educator
01:53

Problem 29

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:29

Problem 29

Find the indicated term in each arithmetic sequence.
$$
80 \text { th term of } 2, \frac{5}{2}, 3, \frac{7}{2}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
00:38

Problem 29

Find the indicated term of each geometric sequence.
$$
\text { 9th term of } 1,-1,1, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
05:27

Problem 29

Show that the formula
$$
2+4+6+\cdots+2 n=n^2+n+2
$$
obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some $k$, it is also true for $k+1$. Then show that the formula is false for $n=1$ (or for any other choice of $n$ ).

Sirat Shah
Sirat Shah
Numerade Educator
01:34

Problem 29

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^6$ in the expansion of $(x+3)^{10}$

John Vanschoick
John Vanschoick
Numerade Educator
01:10

Problem 30

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:35

Problem 30

Find the indicated term in each arithmetic sequence.
$$
70 \text { th term of } 2 \sqrt{5}, 4 \sqrt{5}, 6 \sqrt{5}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
01:20

Problem 30

Find the indicated term of each geometric sequence.
$$
10 \text { th term of }-1,2,-4, \ldots
$$

Dale Sanford
Dale Sanford
Numerade Educator
06:39

Problem 30

Use mathematical induction to prove that if $r \neq 1$, then
$$
a+a r+a r^2+\cdots+a r^{n-1}=a \frac{1-r^n}{1-r}
$$

Sirat Shah
Sirat Shah
Numerade Educator
01:45

Problem 30

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^3$ in the expansion of $(x-3)^{10}$

John Vanschoick
John Vanschoick
Numerade Educator
01:11

Problem 31

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
1,-1,1,-1,1,-1, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
03:58

Problem 31

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
8 th term is $8 ; 20$ th term is 44

Julie Silva
Julie Silva
Numerade Educator
01:01

Problem 31

Find the indicated term of each geometric sequence.
$$
8 \text { th term of } 0.4,0.04,0.004, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
07:27

Problem 31

Use mathematical induction to prove that
$$
\begin{aligned}
& a+(a+d)+(a+2 d) \\
& \quad+\cdots+[a+(n-1) d]=n a+d \frac{n(n-1)}{2}
\end{aligned}
$$

Sirat Shah
Sirat Shah
Numerade Educator
01:23

Problem 31

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^7$ in the expansion of $(2 x-1)^{12}$

Amy Jiang
Amy Jiang
Numerade Educator
01:52

Problem 32

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
03:57

Problem 32

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
4 \text { th term is } 3 ; 20 \text { th term is } 35
$$

Julie Silva
Julie Silva
Numerade Educator
00:44

Problem 32

Find the indicated term of each geometric sequence.
$$
7 \text { th term of } 0.1,1.0,10.0, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
05:02

Problem 32

Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions I and II hold, that is,
(I) A statement is true for a natural number $j$.
(II) If the statement is true for some natural number $k \geq j$, then it is also true for the next natural number $k+1$. then the statement is true for all natural numbers $\geq j$. Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of $n$ sides is $\frac{1}{2} n(n-3)$.

Sirat Shah
Sirat Shah
Numerade Educator
01:16

Problem 32

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^3$ in the expansion of $(2 x+1)^{12}$

Amy Jiang
Amy Jiang
Numerade Educator
01:34

Problem 33

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
1,-2,3,-4,5,-6, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
03:41

Problem 33

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
9 \text { th term is }-5 ; 15 \text { th term is } 31
$$

Julie Silva
Julie Silva
Numerade Educator
00:39

Problem 33

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
7,14,28,56, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:32

Problem 33

Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of $n$ sides equals $(n-2) \cdot 180^{\circ}$.

Sirat Shah
Sirat Shah
Numerade Educator
01:08

Problem 33

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^7$ in the expansion of $(2 x+3)^9$

Amy Jiang
Amy Jiang
Numerade Educator
02:04

Problem 34

The given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$$
2,-4,6,-8,10, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
03:30

Problem 34

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
8 \text { th term is } 4 ; 18 \text { th term is }-96
$$

Julie Silva
Julie Silva
Numerade Educator
00:35

Problem 34

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
5,10,20,40, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
06:55

Problem 34

How would you explain the Principle of Mathematical Induction to a friend?

Sirat Shah
Sirat Shah
Numerade Educator
01:06

Problem 34

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^2$ in the expansion of $(2 x-3)^9$

Amy Jiang
Amy Jiang
Numerade Educator
01:37

Problem 35

A sequence is defined recursively. Write down the first five terms.
$$
a_1=2 ; \quad a_n=3+a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
03:37

Problem 35

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
15 \text { th term is } 0 ; \quad 40 \text { th term is }-50
$$

Julie Silva
Julie Silva
Numerade Educator
01:16

Problem 35

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
-3,1,-\frac{1}{3}, \frac{1}{9}, \ldots
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:18

Problem 35

Solve: $\log _2 \sqrt{x+5}=4$

Sirat Shah
Sirat Shah
Numerade Educator
01:56

Problem 35

Use the Binomial Theorem to find the indicated coefficient or term.
The 5th term in the expansion of $(x+3)^7$

John Vanschoick
John Vanschoick
Numerade Educator
01:38

Problem 36

A sequence is defined recursively. Write down the first five terms.
$$
a_1=3 ; \quad a_n=4-a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
03:31

Problem 36

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
5 \text { th term is }-2 ; 13 \text { th term is } 30
$$

Julie Silva
Julie Silva
Numerade Educator
00:35

Problem 36

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
4,1, \frac{1}{4}, \frac{1}{16}, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
05:07

Problem 36

A mass of $500 \mathrm{~kg}$ is suspended from two cables, as shown in the figure. What are the tensions in the two cables?

Sirat Shah
Sirat Shah
Numerade Educator
00:35

Problem 36

Use the Binomial Theorem to find the indicated coefficient or term.
The 3rd term in the expansion of $(x-3)^7$

Amy Jiang
Amy Jiang
Numerade Educator
01:20

Problem 37

A sequence is defined recursively. Write down the first five terms.
$$
a_1=-2 ; \quad a_n=n+a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
03:38

Problem 37

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
14 \text { th term is }-1 ; \quad 18 \text { th term is }-9
$$

Julie Silva
Julie Silva
Numerade Educator
01:00

Problem 37

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
a_6=243 ; \quad r=-3
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:03

Problem 37

Solve the system: $\left\{\begin{array}{l}4 x+3 y=-7 \\ 2 x-5 y=16\end{array}\right.$

Sirat Shah
Sirat Shah
Numerade Educator
00:33

Problem 37

Use the Binomial Theorem to find the indicated coefficient or term.
The 3 rd term in the expansion of $(3 x-2)^9$

Amy Jiang
Amy Jiang
Numerade Educator
01:22

Problem 38

A sequence is defined recursively. Write down the first five terms.
$$
a_1=1 ; \quad a_n=n-a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
03:30

Problem 38

Find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term.
$$
12 \text { th term is } 4 ; 18 \text { th term is } 28
$$

Julie Silva
Julie Silva
Numerade Educator
00:57

Problem 38

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
a_2=7 ; \quad r=\frac{1}{3}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:46

Problem 38

For $A=\left[\begin{array}{rrr}1 & 2 & -1 \\ 0 & 1 & 4\end{array}\right]$ and $B=\left[\begin{array}{rr}3 & -1 \\ 1 & 0 \\ -2 & 2\end{array}\right]$, find $A \cdot B$.

Sirat Shah
Sirat Shah
Numerade Educator
02:16

Problem 38

Use the Binomial Theorem to find the indicated coefficient or term.
The 6 th term in the expansion of $(3 x+2)^8$

John Vanschoick
John Vanschoick
Numerade Educator
01:11

Problem 39

A sequence is defined recursively. Write down the first five terms.
$$
a_1=5 ; \quad a_n=2 a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
01:03

Problem 39

Find each sum.
$$
1+3+5+\cdots+(2 n-1)
$$

Julie Silva
Julie Silva
Numerade Educator
02:28

Problem 39

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
a_2=7 ; \quad a_4=1575
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:35

Problem 39

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^0$ in the expansion of $\left(x^2+\frac{1}{x}\right)^{12}$

John Vanschoick
John Vanschoick
Numerade Educator
01:07

Problem 40

A sequence is defined recursively. Write down the first five terms.
$$
a_1=2 ; \quad a_n=-a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
00:57

Problem 40

Find each sum.
$$
2+4+6+\cdots+2 n
$$

Julie Silva
Julie Silva
Numerade Educator
03:04

Problem 40

Find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$$
a_3=\frac{1}{3} ; \quad a_6=\frac{1}{81}
$$

Dale Sanford
Dale Sanford
Numerade Educator
View

Problem 40

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^0$ in the expansion of $\left(x-\frac{1}{x^2}\right)^9$

John Vanschoick
John Vanschoick
Numerade Educator
02:17

Problem 41

A sequence is defined recursively. Write down the first five terms.
$$
a_1=3 ; \quad a_n=\frac{a_{n-1}}{n}
$$

Julie Silva
Julie Silva
Numerade Educator
01:06

Problem 41

Find each sum.
$$
7+12+17+\cdots+(2+5 n)
$$

Julie Silva
Julie Silva
Numerade Educator
01:55

Problem 41

Find each sum.
$$
\frac{1}{4}+\frac{2}{4}+\frac{2^2}{4}+\frac{2^3}{4}+\cdots+\frac{2^{n-1}}{4}
$$

Dale Sanford
Dale Sanford
Numerade Educator
04:23

Problem 41

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^4$ in the expansion of $\left(x-\frac{2}{\sqrt{x}}\right)^{10}$

John Vanschoick
John Vanschoick
Numerade Educator
01:23

Problem 42

A sequence is defined recursively. Write down the first five terms.
$$
a_1=-2 ; \quad a_n=n+3 a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
01:08

Problem 42

Find each sum.
$$
-1+3+7+\cdots+(4 n-5)
$$

Julie Silva
Julie Silva
Numerade Educator
01:52

Problem 42

Find each sum.
$$
\frac{3}{9}+\frac{3^2}{9}+\frac{3^3}{9}+\cdots+\frac{3^n}{9}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
04:21

Problem 42

Use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^2$ in the expansion of $\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right)^8$

John Vanschoick
John Vanschoick
Numerade Educator
01:25

Problem 43

A sequence is defined recursively. Write down the first five terms.
$$
a_1=1 ; \quad a_2=2 ; \quad a_n=a_{n-1} \cdot a_{n-2}
$$

Julie Silva
Julie Silva
Numerade Educator
01:56

Problem 43

Find each sum.
$$
2+4+6+\cdots+70
$$

Julie Silva
Julie Silva
Numerade Educator
00:59

Problem 43

Find each sum.
$$
\sum_{k=1}^n\left(\frac{2}{3}\right)^k
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
06:17

Problem 43

Use the Binomial Theorem to find the numerical value of $(1.001)^5$ correct to five decimal places.

John Vanschoick
John Vanschoick
Numerade Educator
01:22

Problem 44

A sequence is defined recursively. Write down the first five terms.
$$
a_1=-1 ; \quad a_2=1 ; \quad a_n=a_{n-2}+n a_{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
01:46

Problem 44

Find each sum.
$$
1+3+5+\cdots+59
$$

Julie Silva
Julie Silva
Numerade Educator
01:47

Problem 44

Find each sum.
$$
\sum_{k=1}^n 4 \cdot 3^{k-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
05:46

Problem 44

Use the Binomial Theorem to find the numerical value of $(0.998)^6$ correct to five decimal places.

John Vanschoick
John Vanschoick
Numerade Educator
01:39

Problem 45

A sequence is defined recursively. Write down the first five terms.
$$
a_1=A ; \quad a_n=a_{n-1}+d
$$

Julie Silva
Julie Silva
Numerade Educator
01:49

Problem 45

Find each sum.
$$
5+9+13+\cdots+49
$$

Julie Silva
Julie Silva
Numerade Educator
01:32

Problem 45

Find each sum.
$$
-1-2-4-8-\cdots-\left(2^{n-1}\right)
$$

Dale Sanford
Dale Sanford
Numerade Educator
00:36

Problem 45

Show that $\left(\begin{array}{c}n \\ n-1\end{array}\right)=n$ and $\left(\begin{array}{l}n \\ n\end{array}\right)=1$.

Amy Jiang
Amy Jiang
Numerade Educator
01:36

Problem 46

A sequence is defined recursively. Write down the first five terms.
$$
a_1=A ; \quad a_n=r a_{n-1}, \quad r \neq 0
$$

Julie Silva
Julie Silva
Numerade Educator
01:49

Problem 46

Find each sum.
$$
2+5+8+\cdots+41
$$

Julie Silva
Julie Silva
Numerade Educator
01:48

Problem 46

Find each sum.
$$
2+\frac{6}{5}+\frac{18}{25}+\cdots+2\left(\frac{3}{5}\right)^{n-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:14

Problem 46

Show that if $n$ and $j$ are integers with $0 \leq j \leq n$, then,
$$
\left(\begin{array}{l}
n \\
j
\end{array}\right)=\left(\begin{array}{c}
n \\
n-j
\end{array}\right)
$$
Conclude that the Pascal triangle is symmetric with respect to a vertical line drawn from the topmost entry.

John Vanschoick
John Vanschoick
Numerade Educator
01:45

Problem 47

A sequence is defined recursively. Write down the first five terms.
$$
a_1=\sqrt{2} ; \quad a_n=\sqrt{2+a_{n-1}}
$$

Julie Silva
Julie Silva
Numerade Educator
02:13

Problem 47

Find each sum.
$$
73+78+83+88+\cdots+558
$$

Julie Silva
Julie Silva
Numerade Educator
01:01

Problem 47

Use a graphing utility to find the sum of each geometric sequence.
$$
\frac{1}{4}+\frac{2}{4}+\frac{2^2}{4}+\frac{2^3}{4}+\cdots+\frac{2^{14}}{4}
$$

Dale Sanford
Dale Sanford
Numerade Educator
05:07

Problem 47

If $n$ is a positive integer, show that
$$
\left(\begin{array}{l}
n \\
0
\end{array}\right)+\left(\begin{array}{l}
n \\
1
\end{array}\right)+\cdots+\left(\begin{array}{l}
n \\
n
\end{array}\right)=2^n
$$

Grant Mansfield
Grant Mansfield
Numerade Educator
02:20

Problem 48

A sequence is defined recursively. Write down the first five terms.
$$
a_1=\sqrt{2} ; \quad a_n=\sqrt{\frac{a_{n-1}}{2}}
$$

Julie Silva
Julie Silva
Numerade Educator
02:18

Problem 48

Find each sum.
$$
7+1-5-11-\cdots-299
$$

Julie Silva
Julie Silva
Numerade Educator
00:56

Problem 48

Use a graphing utility to find the sum of each geometric sequence.
$$
\frac{3}{9}+\frac{3^2}{9}+\frac{3^3}{9}+\cdots+\frac{3^{15}}{9}
$$

Dale Sanford
Dale Sanford
Numerade Educator
04:05

Problem 48

If $n$ is a positive integer, show that
$$
\left(\begin{array}{l}
n \\
0
\end{array}\right)-\left(\begin{array}{l}
n \\
1
\end{array}\right)+\left(\begin{array}{l}
n \\
2
\end{array}\right)-\cdots+(-1)^n\left(\begin{array}{l}
n \\
n
\end{array}\right)=0
$$

Grant Mansfield
Grant Mansfield
Numerade Educator
00:46

Problem 49

Write out each sum.
$$
\sum_{k=1}^n(k+2)
$$

Julie Silva
Julie Silva
Numerade Educator
02:12

Problem 49

Find each sum.
$$
4+4.5+5+5.5+\cdots+100
$$

Julie Silva
Julie Silva
Numerade Educator
00:48

Problem 49

Use a graphing utility to find the sum of each geometric sequence.
$$
\sum_{n=1}^{15}\left(\frac{2}{3}\right)^n
$$

Dale Sanford
Dale Sanford
Numerade Educator
05:29

Problem 49

$\begin{aligned} & \left(\begin{array}{l}5 \\ 0\end{array}\right)\left(\frac{1}{4}\right)^5+\left(\begin{array}{l}5 \\ 1\end{array}\right)\left(\frac{1}{4}\right)^4\left(\frac{3}{4}\right)+\left(\begin{array}{l}5 \\ 2\end{array}\right)\left(\frac{1}{4}\right)^3\left(\frac{3}{4}\right)^2 \\ & +\left(\begin{array}{l}5 \\ 3\end{array}\right)\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3+\left(\begin{array}{l}5 \\ 4\end{array}\right)\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^4+\left(\begin{array}{l}5 \\ 5\end{array}\right)\left(\frac{3}{4}\right)^5=?\end{aligned}$

Grant Mansfield
Grant Mansfield
Numerade Educator
00:52

Problem 50

Write out each sum.
$$
\sum_{k=1}^n(2 k+1)
$$

Julie Silva
Julie Silva
Numerade Educator
02:19

Problem 50

Find each sum.
$$
8+8 \frac{1}{4}+8 \frac{1}{2}+8 \frac{3}{4}+9+\cdots+50
$$

Julie Silva
Julie Silva
Numerade Educator
00:38

Problem 50

Use a graphing utility to find the sum of each geometric sequence.
$$
\sum_{n=1}^{15} 4 \cdot 3^{n-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:01

Problem 50

An approximation for $n$ !, when $n$ is large, is given by
$$
n ! \approx \sqrt{2 n \pi}\left(\frac{n}{e}\right)^n\left(1+\frac{1}{12 n-1}\right)
$$
Calculate 12 !, 20 !, and 25 ! on your calculator. Then use Stirling's formula to approximate 12 !, 20 !, and 25 !.

John Vanschoick
John Vanschoick
Numerade Educator
00:48

Problem 51

Write out each sum.
$$
\sum_{k=1}^n \frac{k^2}{2}
$$

Julie Silva
Julie Silva
Numerade Educator
01:29

Problem 51

Find each sum.
$$
\sum_{n=1}^{80}(2 n-5)
$$

Julie Silva
Julie Silva
Numerade Educator
01:18

Problem 51

Use a graphing utility to find the sum of each geometric sequence.
$$
-1-2-4-8-\cdots-2^{14}
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:18

Problem 51

Solve $6^x=5^{x+1}$. Express the answer both in exact form and as a decimal rounded to three decimal places.

Steven Clarke
Steven Clarke
Numerade Educator
00:50

Problem 52

Write out each sum.
$$
\sum_{k=1}^n(k+1)^2
$$

Julie Silva
Julie Silva
Numerade Educator
01:38

Problem 52

Find each sum.
$$
\sum_{n=1}^{90}(3-2 n)
$$

Julie Silva
Julie Silva
Numerade Educator
00:45

Problem 52

Use a graphing utility to find the sum of each geometric sequence.
$$
2+\frac{6}{5}+\frac{18}{25}+\cdots+2\left(\frac{3}{5}\right)^{15}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:16

Problem 52

For $\mathbf{v}=2 \mathbf{i}+3 \mathbf{j}$ and $\mathbf{w}=3 \mathbf{i}-2 \mathbf{j}$ :
(a) Find the dot product $\mathbf{v} \cdot \mathbf{w}$.
(b) Find the angle between $\mathbf{v}$ and $\mathbf{w}$.
(c) Are the vectors parallel, orthogonal, or neither?

Steven Clarke
Steven Clarke
Numerade Educator
00:53

Problem 53

Write out each sum.
$$
\sum_{k=0}^n \frac{1}{3^k}
$$

Julie Silva
Julie Silva
Numerade Educator
01:40

Problem 53

Find each sum.
$$
\sum_{n=1}^{100}\left(6-\frac{1}{2} n\right)
$$

Julie Silva
Julie Silva
Numerade Educator
01:37

Problem 53

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
1+\frac{1}{3}+\frac{1}{9}+\cdots
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:52

Problem 53

Solve the system of equations:
$$
\left\{\begin{aligned}
x-y-z & =0 \\
2 x+y+3 z & =-1 \\
4 x+2 y-z & =12
\end{aligned}\right.
$$

Suzanne Harwood
Suzanne Harwood
Numerade Educator
00:49

Problem 54

Write out each sum.
$$
\sum_{k=0}^n\left(\frac{3}{2}\right)^k
$$

Julie Silva
Julie Silva
Numerade Educator
02:19

Problem 54

Find each sum.
$$
\sum_{n=1}^{80}\left(\frac{1}{3} n+\frac{1}{2}\right)
$$

Julie Silva
Julie Silva
Numerade Educator
01:23

Problem 54

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
2+\frac{4}{3}+\frac{8}{9}+\cdots
$$

Dale Sanford
Dale Sanford
Numerade Educator
03:25

Problem 54

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
$$
\left\{\begin{array}{r}
x \geq 0 \\
y \geq 0 \\
x+y \leq 6 \\
2 x+y \leq 10
\end{array}\right.
$$

Maria Dearborn
Maria Dearborn
Numerade Educator
01:24

Problem 55

Write out each sum.
$$
\sum_{k=0}^{n-1} \frac{1}{3^{k+1}}
$$

Julie Silva
Julie Silva
Numerade Educator
01:35

Problem 55

Find each sum.
The sum of the first 120 terms of the sequence $14,16,18,20, \ldots$

Julie Silva
Julie Silva
Numerade Educator
01:34

Problem 55

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
8+4+2+\cdots
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:24

Problem 56

Write out each sum.
$$
\sum_{k=0}^{n-1}(2 k+1)
$$

Julie Silva
Julie Silva
Numerade Educator
01:33

Problem 56

Find each sum.
The sum of the first 46 terms of the sequence
$$
2,-1,-4,-7, \ldots
$$

Julie Silva
Julie Silva
Numerade Educator
00:54

Problem 56

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
6+2+\frac{2}{3}+\cdots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:51

Problem 57

Write out each sum.
$$
\sum_{k=2}^n(-1)^k \ln k
$$

Madysn Cardinal
Madysn Cardinal
Numerade Educator
02:01

Problem 57

Find $x$ so that $x+3,2 x+1$, and $5 x+2$ are consecutive terms of an arithmetic sequence.

Julie Silva
Julie Silva
Numerade Educator
01:06

Problem 57

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\cdots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:51

Problem 58

Write out each sum.
$$
\sum_{k=3}^n(-1)^{k+1} 2^k
$$

Julie Silva
Julie Silva
Numerade Educator
01:20

Problem 58

Find $x$ so that $2 x, 3 x+2$, and $5 x+3$ are consecutive terms of an arithmetic sequence.

Julie Silva
Julie Silva
Numerade Educator
01:40

Problem 58

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
1-\frac{3}{4}+\frac{9}{16}-\frac{27}{64}+\cdots
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:08

Problem 59

Express each sum using summation notation.
$$
1+2+3+\cdots+20
$$

Julie Silva
Julie Silva
Numerade Educator
04:26

Problem 59

How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of 1092 ?

Julie Silva
Julie Silva
Numerade Educator
00:28

Problem 59

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
8+12+18+27+\cdots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:36

Problem 60

Express each sum using summation notation.
$$
1^3+2^3+3^3+\cdots+8^3
$$

Julie Silva
Julie Silva
Numerade Educator
04:26

Problem 60

How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of 1092 ?

Julie Silva
Julie Silva
Numerade Educator
01:03

Problem 60

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
9+12+16+\frac{64}{3}+\cdots
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:25

Problem 61

Express each sum using summation notation.
$$
\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{13}{13+1}
$$

Julie Silva
Julie Silva
Numerade Educator
01:40

Problem 61

The Drury Lane Theater has 25 seats in the first row and 30 rows in all. Each successive row contains one additional seat. How many seats are in the theater?

Julie Silva
Julie Silva
Numerade Educator
01:24

Problem 61

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 5\left(\frac{1}{4}\right)^{k-1}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:20

Problem 62

Express each sum using summation notation.
$$
1+3+5+7+\cdots+[2(12)-1]
$$

Julie Silva
Julie Silva
Numerade Educator
01:50

Problem 62

The corner section of a football stadium has 15 seats in the first row and 40 rows in all. Each successive row contains two additional seats. How many seats are in this section?

Julie Silva
Julie Silva
Numerade Educator
01:33

Problem 62

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:48

Problem 63

Express each sum using summation notation.
$$
1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots+(-1)^6\left(\frac{1}{3^6}\right)
$$

Julie Silva
Julie Silva
Numerade Educator
01:59

Problem 63

A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the illustration. How many tiles of each color will be required?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:51

Problem 63

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} \frac{1}{2} \cdot 3^{k-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:33

Problem 64

Express each sum using summation notation.
$$
\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\cdots+(-1)^{12}\left(\frac{2}{3}\right)^{11}
$$

Julie Silva
Julie Silva
Numerade Educator
02:28

Problem 64

A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two fewer bricks than the prior step.
(a) How many bricks are required for the top step?
(b) How many bricks are required to build the staircase?

Julie Silva
Julie Silva
Numerade Educator
00:53

Problem 64

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
00:32

Problem 65

Express each sum using summation notation.
$$
3+\frac{3^2}{2}+\frac{3^3}{3}+\cdots+\frac{3^n}{n}
$$

Julie Silva
Julie Silva
Numerade Educator
01:41

Problem 65

As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of $5.5^{\circ} \mathrm{F}$ per 1000 feet until it reaches its dew point. If the ground temperature is $67^{\circ} \mathrm{F}$, write a formula for the sequence of temperatures, $\left\{T_n\right\}$, of a parcel of air that has risen $n$ thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet?

Julie Silva
Julie Silva
Numerade Educator
01:32

Problem 65

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 6\left(-\frac{2}{3}\right)^{k-1}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:02

Problem 66

Express each sum using summation notation.
$$
\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+\cdots+\frac{n}{e^n}
$$

Julie Silva
Julie Silva
Numerade Educator
02:49

Problem 66

Ladders used by fruit pickers are typically tapered with a wide bottom for stability and a narrow top for ease of picking. If the bottom rung of such a ladder is 49 inches wide and the top rung is 24 inches wide, how many rungs does the ladder have if each rung is 2.5 inches shorter than the one below it? How much material would be needed to make the rungs for the ladder described?

Julie Silva
Julie Silva
Numerade Educator
01:21

Problem 66

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:17

Problem 67

Express each sum using summation notation.
$$
a+(a+d)+(a+2 d)+\cdots+(a+n d)
$$

Julie Silva
Julie Silva
Numerade Educator
01:37

Problem 67

An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Julie Silva
Julie Silva
Numerade Educator
02:48

Problem 67

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 3\left(\frac{2}{3}\right)^k
$$

Dale Sanford
Dale Sanford
Numerade Educator
00:50

Problem 68

Express each sum using summation notation.
$$
a+a r+a r^2+\cdots+a r^{n-1}
$$

Julie Silva
Julie Silva
Numerade Educator
04:22

Problem 68

How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats?

Julie Silva
Julie Silva
Numerade Educator
01:15

Problem 68

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$$
\sum_{k=1}^{\infty} 2\left(\frac{3}{4}\right)^k
$$

Dale Sanford
Dale Sanford
Numerade Educator
00:39

Problem 69

Find the sum of each sequence.
$$
\sum_{k=1}^{40} 5
$$

Julie Silva
Julie Silva
Numerade Educator
04:04

Problem 69

If you take a job with a starting salary of $$\$ 35,000$$ per year and a guaranteed raise of $$\$ 1400$$ per year, how many years will it be before your aggregate salary is $$\$ 280,000$$ ?

Vishal Parmar
Vishal Parmar
Numerade Educator
01:16

Problem 69

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\{n+2\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:32

Problem 70

Find the sum of each sequence.
$$
\sum_{k=1}^{50} 8
$$

Julie Silva
Julie Silva
Numerade Educator
01:01

Problem 70

Make up an arithmetic sequence. Give it to a friend and ask for its 20th term.

Vishal Parmar
Vishal Parmar
Numerade Educator
02:49

Problem 70

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\{2 n-5\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
00:49

Problem 71

Find the sum of each sequence.
$$
\sum_{k=1}^{40} k
$$

Julie Silva
Julie Silva
Numerade Educator
02:15

Problem 71

Describe the similarities and differences between arithmetic sequences and linear functions.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:56

Problem 71

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{4 n^2\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:14

Problem 72

Find the sum of each sequence.
$$
\sum_{k=1}^{24}(-k)
$$

Julie Silva
Julie Silva
Numerade Educator
01:25

Problem 72

If a credit card charges $15.3 \%$ interest compounded monthly, find the effective rate of interest.

Julie Silva
Julie Silva
Numerade Educator
02:05

Problem 72

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{5 n^2+1\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:53

Problem 73

Find the sum of each sequence.
$$
\sum_{k=1}^{20}(5 k+3)
$$

Julie Silva
Julie Silva
Numerade Educator
01:52

Problem 73

The vector $\mathbf{v}$ has initial point $P=(-1,2)$ and terminal point $Q=(3,-4)$. Write $\mathbf{v}$ in the form $a \mathbf{i}+b \mathbf{j}$; that is, find its position vector.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:07

Problem 73

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{3-\frac{2}{3} n\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:24

Problem 74

Find the sum of each sequence.
$$
\sum_{k=1}^{26}(3 k-7)
$$

Julie Silva
Julie Silva
Numerade Educator
03:42

Problem 74

Analyze and graph the equation: $25 x^2+4 y^2=100$

Julie Silva
Julie Silva
Numerade Educator
03:01

Problem 74

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{8-\frac{3}{4} n\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
01:34

Problem 75

Find the sum of each sequence.
$$
\sum_{k=1}^{16}\left(k^2+4\right)
$$

Julie Silva
Julie Silva
Numerade Educator
01:32

Problem 75

Find the inverse of the matrix $\left[\begin{array}{rr}2 & 0 \\ 3 & -1\end{array}\right]$, if there is one; otherwise, state that the matrix is singular.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:46

Problem 75

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
1,3,6,10, \ldots
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:37

Problem 76

Find the sum of each sequence.
$$
\sum_{k=0}^{14}\left(k^2-4\right)
$$

Julie Silva
Julie Silva
Numerade Educator
01:15

Problem 76

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
2,4,6,8, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:14

Problem 77

Find the sum of each sequence.
$$
\sum_{k=10}^{60}(2 k)
$$

Julie Silva
Julie Silva
Numerade Educator
03:38

Problem 77

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{\left(\frac{2}{3}\right)^n\right\}
$$

Dale Sanford
Dale Sanford
Numerade Educator
02:17

Problem 78

Find the sum of each sequence.
$$
\sum_{k=8}^{40}(-3 k)
$$

Julie Silva
Julie Silva
Numerade Educator
01:26

Problem 78

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{\left(\frac{5}{4}\right)^n\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:02

Problem 79

Find the sum of each sequence.
$$
\sum_{k=5}^{20} k^3
$$

Julie Silva
Julie Silva
Numerade Educator
01:27

Problem 79

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
-1,2,-4,8, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
02:03

Problem 80

Find the sum of each sequence.
$$
\sum_{k=4}^{24} k^3
$$

Julie Silva
Julie Silva
Numerade Educator
01:25

Problem 80

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
1,1,2,3,5,8, \ldots
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:39

Problem 81

John has a balance of $$\$ 3000$$ on his Discover card, which charges $1 \%$ interest per month at the time the payment is made on any unpaid balance from the previous month. John can afford to pay $$\$ 100$$ toward the balance each month. His balance each month after making a $$\$ 100$$ payment is given by the recursively defined sequence
$$
B_0=\$ 3000 \quad B_n=1.01 B_{n-1}-100
$$
Determine John's balance after making the first payment. That is, determine $B_1$.

Madysn Cardinal
Madysn Cardinal
Numerade Educator
01:43

Problem 81

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{3^{n / 2}\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:52

Problem 82

A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. It is also known that the trout population is growing at a rate of $3 \%$ per month. The size of the population after $n$ months is given by the recursively defined sequence
$$
p_0=2000 \quad p_n=1.03 p_{n-1}+20
$$
How many trout are in the pond after 2 months? That is, what is $p_2$ ?

Julie Silva
Julie Silva
Numerade Educator
01:17

Problem 82

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms.
$$
\left\{(-1)^n\right\}
$$

Amit Srivastava
Amit Srivastava
Numerade Educator
01:27

Problem 83

Phil bought a car by taking out a loan for $$\$ 18,500$$ at $0.5 \%$ interest per month. Phil's normal monthly payment is $$\$ 434.47$$ per month, but he decides that he can afford to pay $$\$ 100$$ extra toward the balance each month. His balance each month is given by the recursively defined sequence
$$
B_0=18,500 \quad B_n=1.005 B_{n-1}-534.47
$$
Determine Phil's balance after making the first payment. That is, determine $B_1$.

Julie Silva
Julie Silva
Numerade Educator
00:55

Problem 83

Find $x$ so that $x, x+2$, and $x+3$ are consecutive terms of a geometric sequence.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:33

Problem 84

The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that $10 \%$ of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after $n$ years is given by the recursively defined sequence
$$
p_0=250 \quad p_n=0.9 p_{n-1}+15
$$
Determine the amount of pollutant in the lake after 2 years. That is, determine $p_2$.

Julie Silva
Julie Silva
Numerade Educator
01:11

Problem 84

Find $x$ so that $x-1, x$, and $x+2$ are consecutive terms of a geometric sequence.

Dale Sanford
Dale Sanford
Numerade Educator

Problem 85

A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months? See illustration, top right.

Check back soon!
01:43

Problem 85

If you have been hired at an annual salary of $$\$ 42,000$$ and expect to receive annual increases of $3 \%$, what will your salary be when you begin your fifth year?

Dale Sanford
Dale Sanford
Numerade Educator
06:08

Problem 86

Let
$$
u_n=\frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}
$$
define the $n$th term of a sequence.
(a) Show that $u_1=1$ and $u_2=1$.
(b) Show that $u_{n+2}=u_{n+1}+u_n$.
(c) Draw the conclusion that $\left\{u_n\right\}$ is a Fibonacci sequence.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:44

Problem 86

A new piece of equipment cost a company $$\$ 15,000$$. Each year, for tax purposes, the company depreciates the value by $15 \%$. What value should the company give the equipment after 5 years?

Dale Sanford
Dale Sanford
Numerade Educator
01:11

Problem 87

Divide the triangular array shown (called the Pascal triangle) using diagonal lines as indicated. Find the sum of the numbers in each diagonal row. Do you recognize this sequence?

Julie Silva
Julie Silva
Numerade Educator
02:56

Problem 87

Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length.
(a) What is the length of the arc of the 10th swing?
(b) On which swing is the length of the arc first less than 1 foot?
(c) After 15 swings, what total length will the pendulum have swung?
(d) When it stops, what total length will the pendulum have swung?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:52

Problem 88

Use the result of Problem 86 to do the following problems.
(a) Write the first 11 terms of the Fibonacci sequence.
(b) Write the first 10 terms of the ratio $\frac{u_{n+1}}{u_n}$.
(c) As $n$ gets large, what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the façade of the Parthenon was constructed using the golden ratio.
(d) Write down the first 10 terms of the ratio $\frac{u_n}{u_{n+1}}$.
(e) As $n$ gets large, what number does the ratio approach? This number is referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.

Anurag Kumar
Anurag Kumar
Numerade Educator
04:43

Problem 88

A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.
(a) What height will the ball bounce up to after it strikes the ground for the third time?
(b) How high will it bounce after it strikes the ground for the $n$th time?
(c) How many times does the ball need to strike the ground before its bounce is less than 6 inches?
(d) What total vertical distance does the ball travel before it stops bouncing?

Amit Srivastava
Amit Srivastava
Numerade Educator
03:12

Problem 89

Approximating $f(x)=e^x$ In calculus, it can be shown that
$$
f(x)=e^x=\sum_{k=0}^{\infty} \frac{x^k}{k !}
$$
We can approximate the value of $f(x)=e^x$ for any $x$ using the following sum
$$
f(x)=e^x \approx \sum_{k=0}^n \frac{x^k}{k !}
$$
for some $n$.
(a) Approximate $f(1.3)$ with $n=4$
(b) Approximate $f(1.3)$ with $n=7$.
(c) Use a calculator to approximate $f(1.3)$.
(d) Using trial and error, along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(1.3)$ correct to eight decimal places.

Anurag Kumar
Anurag Kumar
Numerade Educator
02:03

Problem 89

Christine contributes $$\$ 100$$ each month to her 401(k). What will be the value of Christine's 401(k) after the 360 th deposit (30 years) if the per annum rate of return is assumed to be $12 \%$ compounded monthly?

Dale Sanford
Dale Sanford
Numerade Educator
02:26

Problem 90

Approximating $f(x)=e^x$ Refer to Problem 89 .
(a) Approximate $f(-2.4)$ with $n=3$.
(b) Approximate $f(-2.4)$ with $n=6$.
(c) Use a calculator to approximate $f(-2.4)$.
(d) Using trial and error, along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(-2.4)$ correct to eight decimal places.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:55

Problem 90

Jolene wants to purchase a new home. Suppose that she invests $$\$ 400$$ per month into a mutual fund. If the per annum rate of return of the mutual fund is assumed to be $10 \%$ compounded monthly, how much will Jolene have for a down payment after the 36 th deposit (3 years)?

Vishal Parmar
Vishal Parmar
Numerade Educator
06:48

Problem 91

Bode's Law In 1772, Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun:
$$
a_1=0.4 \quad a_n=0.4+0.3 \cdot 2^{n-2}
$$
where $n \geq 2$ is the number of the planet from the sun.
(a) Determine the first eight terms of this sequence.
(b) At the time of Bode's publication, the known planets were Mercury (0.39 AU), Venus ( $0.72 \mathrm{AU})$, Earth (1 AU), Mars (1.52 AU), Jupiter (5.20 AU), and Saturn (9.54 AU). How do the actual distances compare to the terms of the sequence?
(c) The planet Uranus was discovered in 1781 , and the asteroid Ceres was discovered in 1801. The mean orbital distances from the sun to Uranus and Ceres* are 19.2 AU and 2.77 AU, respectively. How well do these values fit within the sequence?
(d) Determine the ninth and tenth terms of Bode's sequence.
(e) The planets Neptune and Pluto* were discovered in 1846 and 1930, respectively. Their mean orbital distances from the sun are 30.07 AU and 39.44 AU, respectively. How do these actual distances compare to the terms of the sequence?
(f) On July 29, 2005, NASA announced the discovery of a dwarf planet* $(n=11)$, which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:40

Problem 91

Don contributes $$\$ 500$$ at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80th deposit (20 years) if the per annum rate of return is assumed to be $8 \%$ compounded quarterly?

Dale Sanford
Dale Sanford
Numerade Educator
01:05

Problem 92

Show that
$$
1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2}
$$
[Hint: Let
$$
\begin{aligned}
& S=1+2+\cdots+(n-1)+n \\
& S=n+(n-1)+(n-2)+\cdots+1
\end{aligned}
$$
Add these equations. Then
$$
2 S=[1+n]+[2+(n-1)]+\cdots+[n+1]
$$
$n$ terms in bracket
Now complete the derivation.]

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 92

Ray contributes $$\$ 1000$$ to an individual retirement account (IRA) semiannually. What will the value of the IRA be when Ray makes his 30th deposit (after 15 years) if the per annum rate of return is assumed to be $10 \%$ compounded semiannually?

Dale Sanford
Dale Sanford
Numerade Educator
02:45

Problem 93

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$
a_0=k \quad a_n=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)
$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_5$. Compare this result to the value provided by your calculator.
$$
\sqrt{5}
$$

Yujie Wang
Yujie Wang
College of San Mateo
02:17

Problem 93

Scott and Alice want to purchase a vacation home in 10 years and need $$\$ 50,000$$ for a down payment. How much should they place in a savings account each month if the per annum rate of return is assumed to be $6 \%$ compounded monthly?

Vishal Parmar
Vishal Parmar
Numerade Educator
02:26

Problem 94

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$
a_0=k \quad a_n=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)
$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_5$. Compare this result to the value provided by your calculator.
$$
\sqrt{8}
$$

Yujie Wang
Yujie Wang
College of San Mateo
03:05

Problem 94

For a child born in 2017, the cost of a 4-year college education at a public university is projected to be $$\$ 185,000$$. Assuming an $8 \%$ per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have $$\$ 185,000$$ in 18 years when the child begins college?

Yujie Wang
Yujie Wang
College of San Mateo
07:38

Problem 95

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$
a_0=k \quad a_n=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)
$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_5$. Compare this result to the value provided by your calculator.
$$
\sqrt{21}
$$

Andrew Sum
Andrew Sum
Numerade Educator
02:19

Problem 95

In an old fable, a commoner who had saved the king's life was told he could ask the king for any just reward. Being a shrewd man, the commoner said, "A simple wish, sire. Place one grain of wheat on the first square of a chessboard, two grains on the second square, four grains on the third square, continuing until you have filled the board. This is all I seek." Compute the total number of grains needed to do this to see why the request, seemingly simple, could not be granted. (A chessboard consists of $8 \times 8=64$ squares.)

Dale Sanford
Dale Sanford
Numerade Educator
02:05

Problem 96

A method for approximating $\sqrt{p}$ can be traced back to the Babylonians. The formula is given by the recursively defined sequence
$$
a_0=k \quad a_n=\frac{1}{2}\left(a_{n-1}+\frac{p}{a_{n-1}}\right)
$$
where $k$ is an initial guess as to the value of the square root. Use this recursive formula to approximate the following square roots by finding $a_5$. Compare this result to the value provided by your calculator.
$$
\sqrt{89}
$$

Yujie Wang
Yujie Wang
College of San Mateo
02:35

Problem 96

Look at the figure. What fraction of the square is eventually shaded if the indicated shading process continues indefinitely?

Yujie Wang
Yujie Wang
College of San Mateo
02:16

Problem 97

Triangular Numbers A triangular number is a term of the sequence
$$
u_1=1 \quad u_{n+1}=u_n+(n+1)
$$
Write down the first seven triangular numbers.

Julie Silva
Julie Silva
Numerade Educator
01:20

Problem 97

Suppose that, throughout the U.S. economy, individuals spend $90 \%$ of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is 0.90 . For example, if Jane earns an additional dollar, she will spend $$0.9(1)=\$ 0.90$$ of it. The individual who earns $$\$ 0.90$$ (from Jane) will spend $90 \%$ of it, or $$\$ 0.81$$. This process of spending continues and results in an infinite geometric series as follows:
$$
1,0.90,0.90^2, 0.90^3, 0.90^4, \ldots
$$
The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend $90 \%$ of every additional dollar that they earn?

Dale Sanford
Dale Sanford
Numerade Educator
02:05

Problem 98

For the sequence given in Problem 97, show that
$$
u_{n+1}=\frac{(n+1)(n+2)}{2} .
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:25

Problem 98

Refer to Problem 97. Suppose that the marginal propensity to consume throughout the U.S. economy is 0.95 . What is the multiplier for the U.S. economy?

Dale Sanford
Dale Sanford
Numerade Educator
01:13

Problem 99

For the sequence given in Problem 97, show that
$$
u_{n+1}+u_n=(n+1)^2
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:08

Problem 99

One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays $$\$ P$$ per year in dividends, and historically, the dividend has been increased $i \%$ per year. If you desire an annual rate of return of $r \%$, this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments:
$$
\text { Price }=P+P \cdot \frac{1+i}{1+r}+P \cdot\left(\frac{1+i}{1+r}\right)^2+P \cdot\left(\frac{1+i}{1+r}\right)^3+\cdots
$$
The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of $$\$ 4.00$$, and historically, the dividend has been increased $3 \%$ per year. You desire an annual rate of return of $9 \%$. What is the most you should pay for the stock?

Dale Sanford
Dale Sanford
Numerade Educator
01:38

Problem 100

Investigate various applications that lead to a Fibonacci sequence, such as in art, architecture, or financial markets. Write an essay on these applications.

Julie Silva
Julie Silva
Numerade Educator
01:42

Problem 100

Refer to Problem 99. Suppose that a stock pays an annual dividend of $$\$ 2.50$$, and historically, the dividend has increased $4 \%$ per year. You desire an annual rate of return of $11 \%$. What is the most that you should pay for the stock?

Dale Sanford
Dale Sanford
Numerade Educator
01:10

Problem 101

Write a paragraph that explains why the numbers found in Problem 97 are called triangular.

Julie Silva
Julie Silva
Numerade Educator
02:09

Problem 101

A rich man promises to give you $$\$ 1000$$ on September 1,2017. Each day thereafter he will give you $\frac{9}{10}$ of what he gave you the previous day. What is the first date on which the amount you receive is less than $1 \phi$ ? How much have you received when this happens?

Aman Gupta
Aman Gupta
Numerade Educator
01:52

Problem 102

If $$\$ 2500$$ is invested at $3 \%$ compounded monthly, find the amount that results after a period of 2 years.

Julie Silva
Julie Silva
Numerade Educator
05:22

Problem 102

A special section in the end zone of a football stadium has 2 seats in the first row and 14 rows total. Each successive row has 2 seats more than the row before. In this particular section, the first seat is sold for 1 cent, and each following seat sells for $5 \%$ more than the previous seat. Find the total revenue generated if every seat in the section is sold. Round only the final answer, and state the final answer in dollars rounded to two decimal places.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:32

Problem 103

Write the complex number $-1-i$ in polar form. Express the argument in degrees.

Lauren Shelton
Lauren Shelton
Numerade Educator
01:48

Problem 103

You are offered two jobs. Job A has a starting salary of $$\$ 50,000$$ with annual raises of $$\$ 2,000$$. Job B has a starting salary of $$\$ 48,000$$ with annual raises of $3 \%$. Find how many years it will take for the cumulative salary of job B to equal that of A. State the answer in years rounded to one decimal place. $(\mathrm{JJC})^{\dagger}$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
00:55

Problem 104

For $\mathbf{v}=2 \mathbf{i}-\mathbf{j}+3 \mathbf{k}$ and $\mathbf{w}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}$, find the cross product $\mathbf{v} \times \mathbf{w}$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
22:29

Problem 104

A fractal known as the Koch Curve is generated by beginning with an equilateral triangle and then adding more triangles that are similar to the original, as pictured below. This process continues forever. (JJC) ${ }^{\dagger}$ Computing the total eventual area by adding each additional area onto the original gives the following series. Evaluate and state the exact answer in square meters. $$
A=2+2 \cdot 3\left(\frac{1}{9}\right)+2 \cdot 12\left(\frac{1}{9}\right)^2+2 \cdot 48\left(\frac{1}{9}\right)^3+2 \cdot 192\left(\frac{1}{9}\right)^4+\ldots
$$
${ }^{\dagger}$ Courtesy of the Joliet Junior College Mathematics Department.

Geena Pullo
Geena Pullo
Numerade Educator
02:10

Problem 105

Find an equation of the parabola with vertex $(-3,4)$ and focus $(1,4)$.

Julie Silva
Julie Silva
Numerade Educator
03:17

Problem 105

Critical Thinking You are interviewing for a job and receive two offers for a five-year contract:
A: $$\$ 40,000$$ to start, with guaranteed annual increases of $6 \%$ for the first 5 years
$$B: \$ 44,000$$ to start, with guaranteed annual increases of $3 \%$ for the first 5 years
Which offer is better if your goal is to be making as much as possible after 5 years? Which is better if your goal is to make as much money as possible over the contract (5 years)?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:14

Problem 106

Critical Thinking Which of the following choices, $A$ or $B$, results in more money?
$A$ : To receive $$\$ 1000$$ on day $$1, \$ 999$$ on day $$2, \$ 998$$ on day 3 , with the process to end after 1000 days
$B$ : To receive $$\$ 1$$ on day $$1, \$ 2$$ on day $$2, \$ 4$$ on day 3 , for 19 days

Dale Sanford
Dale Sanford
Numerade Educator
07:12

Problem 107

You have just signed a 7-year professional football league contract with a beginning salary of $$\$ 2,000,000$$ per year. Management gives you the following options with regard to your salary over the 7 years.
1. A bonus of $$\$ 100,000$$ each year
2. An annual increase of $4.5 \%$ per year beginning after 1 year
3. An annual increase of $$\$ 95,000$$ per year beginning after 1 year
Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

Yujie Wang
Yujie Wang
College of San Mateo
01:37

Problem 108

Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor. Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies. Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much do you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:52

Problem 109

Can a sequence be both arithmetic and geometric? Give reasons for your answer.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:09

Problem 110

Make up a geometric sequence. Give it to a friend and ask for its 20 th term.

Vishal Parmar
Vishal Parmar
Numerade Educator
01:37

Problem 111

Make up two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:54

Problem 112

Describe the similarities and differences between geometric sequences and exponential functions.

Amit Srivastava
Amit Srivastava
Numerade Educator
00:52

Problem 113

Use the Change-of-Base Formula and a calculator to evaluate $\log _7 62$. Round the answer to three decimal places.

Dale Sanford
Dale Sanford
Numerade Educator
01:15

Problem 114

Find the unit vector in the same direction as $\mathbf{v}=8 \mathbf{i}-15 \mathbf{j}$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:42

Problem 115

Find the equation of the hyperbola with vertices at $(-2,0)$ and $(2,0)$ and a focus at $(4,0)$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:26

Problem 116

Find the value of the determinant: $\left|\begin{array}{rrr}3 & 1 & 0 \\ 0 & -2 & 6 \\ 4 & -1 & -2\end{array}\right|$.

Amit Srivastava
Amit Srivastava
Numerade Educator