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Precalculus: pearson new international edition

Michael Sullivan

Chapter 12

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 $a(n)$ ______________ 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

Problem 1

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$2+4+6+\cdots+2 n=n(n+1)$

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01:09

Problem 1

_________ _________is a triangular display of the binomial coefficients.

Grant Mansfield
Grant Mansfield
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

Problem 2

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+5+9+\cdots+(4 n-3)=n(2 n-1)$

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00:43

Problem 2

$\left(\begin{array}{l}n \\ 0\end{array}\right)=$ __________ and $\left(\begin{array}{l}n \\ 1\end{array}\right)=$ ____________

Amy Jiang
Amy Jiang
Numerade Educator
00:17

Problem 3

$A(n)$ __________ is a function whose domain is the set of positive integers.

Julie Silva
Julie Silva
Numerade Educator
00:32

Problem 3

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

Vishal Parmar
Vishal Parmar
Numerade Educator
01:01

Problem 3

In a(n) __________sequence the ratio of successive terms is a constant.

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 3

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$3+4+5+\cdots+(n+2)=\frac{1}{2} n(n+5)$

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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 __________.

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 4

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$3+5+7+\cdots+(2 n+1)=n(n+2)$

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00:45

Problem 4

The __________ __________ can be used to expand expressions like $(2 x+3)^6$.

John Vanschoick
John Vanschoick
Numerade Educator
00:35

Problem 5

If $n \geq 0$ is an integer, then $n !=$ _________ when $n \geq 2$.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:08

Problem 5

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{s_n\right\}=\{n+4\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:13

Problem 5

If a series does not converge, it is called a __________.

Vishal Parmar
Vishal Parmar
Numerade Educator

Problem 5

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$2+5+8+\cdots+(3 n-1)=\frac{1}{2} n(3 n+1)$

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01:21

Problem 5

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}5 \\ 3\end{array}\right)$

Julie Silva
Julie Silva
Numerade Educator
00:19

Problem 6

The sequence $a_1=5, a_n=3 a_{n-1}$ is an example of a __________ sequence.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:20

Problem 6

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{s_n\right\}=\{n-5\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:39

Problem 6

True or False A geometric sequence may be defined recursively.

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 6

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+4+7+\cdots+(3 n-2)=\frac{1}{2} n(3 n-1)$

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Problem 6

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}7 \\ 3\end{array}\right)$

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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
01:08

Problem 7

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{a_n\right\}=\{2 n-5\}$

Raushan Kumar
Raushan Kumar
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

Problem 7

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+2+2^2+\cdots+2^{n-1}=2^n-1$

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00:25

Problem 7

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}7 \\ 5\end{array}\right)$

Amy Jiang
Amy Jiang
Numerade Educator
00:31

Problem 8

True or False $\sum_{k=1}^n k=1+2+3+\cdots+n=\frac{n(n+1)}{2}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:21

Problem 8

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{b_n\right\}=\{3 n+1\}$

Raushan Kumar
Raushan Kumar
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

Problem 8

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+3+3^2+\cdots+3^{n-1}=\frac{1}{2}\left(3^n-1\right)$

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00:32

Problem 8

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}9 \\ 7\end{array}\right)$

Jackie Heldman
Jackie Heldman
Numerade Educator
00:29

Problem 9

In Problems 9-14, evaluate each factorial expression.
10 !

Anurag Kumar
Anurag Kumar
Numerade Educator
01:28

Problem 9

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{c_n\right\}=\{6-2 n\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 9

In Problems 9-18, 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

Problem 9

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+4+4^2+\cdots+4^{n-1}=\frac{1}{3}\left(4^n-1\right)$

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Problem 9

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}50 \\ 49\end{array}\right)$

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00:27

Problem 10

In Problems 9-14, evaluate each factorial expression.
9 !

Anurag Kumar
Anurag Kumar
Numerade Educator
01:17

Problem 10

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{a_n\right\}=\{4-2 n\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 10

In Problems 9-18, 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

Problem 10

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1+5+5^2+\cdots+5^{n-1}=\frac{1}{4}\left(5^n-1\right)$

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Problem 10

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{c}100 \\ 98\end{array}\right)$

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00:22

Problem 11

In Problems 9-14, evaluate each factorial expression.
$\frac{9 !}{6 !}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:56

Problem 11

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{t_n\right\}=\left\{\frac{1}{2}-\frac{1}{3} n\right\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 11

In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$\left|a_n\right|=\left\{-3\left(\frac{1}{2}\right)^n\right\}$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 11

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$

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00:55

Problem 11

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}1000 \\ 1000\end{array}\right)$

John Vanschoick
John Vanschoick
Numerade Educator
00:24

Problem 12

In Problems 9-14, evaluate each factorial expression.
$\frac{12 !}{10 !}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:06

Problem 12

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{t_n\right\}=\left\{\frac{2}{3}+\frac{n}{4}\right\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 12

In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$\left|b_n\right|=\left\{\left(\frac{5}{2}\right)^n\right\}$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 12

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}=\frac{n}{2 n+1}$

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00:18

Problem 12

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{c}1000 \\ 0\end{array}\right)$

Amy Jiang
Amy Jiang
Numerade Educator
00:29

Problem 13

In Problems 9-14, evaluate each factorial expression.
$\frac{3 ! 7 !}{4 !}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:08

Problem 13

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{s_n\right\}=\left\{\ln 3^n\right\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 13

In Problems 9-18, show that each sequence is geometric. Then find the common ratio and write out the first four terms.
$\left\{c_n\right\}=\left\{\frac{2^{n-1}}{4}\right\}$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 13

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1^2+2^2+3^2+\cdots+n^2=\frac{1}{6} n(n+1)(2 n+1)$

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Problem 13

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}55 \\ 23\end{array}\right)$

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00:29

Problem 14

In Problems 9-14, evaluate each factorial expression.
$\frac{5 ! 8 !}{3 !}$

Anurag Kumar
Anurag Kumar
Numerade Educator
00:51

Problem 14

In Problems 5-14, show that each sequence is arithmetic. Find the common difference and write out the first four terms.
$\left\{s_n\right\}=\left\{e^{\ln n}\right\}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 14

In Problems 9-18, 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

Problem 14

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1^3+2^3+3^3+\cdots+n^3=\frac{1}{4} n^2(n+1)^2$

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00:49

Problem 14

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}60 \\ 20\end{array}\right)$

John Vanschoick
John Vanschoick
Numerade Educator
00:26

Problem 15

In Problems 15-26, write down the first five terms of each sequence.
$\left\{s_n\right\}=\{n\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:28

Problem 15

In Problems 15-22, 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 fifty-first term?
$a_1=2 ; d=3$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 15

In Problems 9-18, 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

Problem 15

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$4+3+2+\cdots+(5-n)=\frac{1}{2} n(9-n)$

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Problem 15

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}47 \\ 25\end{array}\right)$

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00:37

Problem 16

In Problems 15-26, write down the first five terms of each sequence.
$\left\{s_n\right\}=\left\{n^2+1\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:22

Problem 16

In Problems 15-22, 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 fifty-first term?
$a_1=-2 ; \quad d=4$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 16

In Problems 9-18, 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

Problem 16

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$-2-3-4-\cdots-(n+1)=-\frac{1}{2} n(n+3)$

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00:52

Problem 16

In Problems 5-16, evaluate each expression.
$\left(\begin{array}{l}37 \\ 19\end{array}\right)$

John Vanschoick
John Vanschoick
Numerade Educator
00:48

Problem 17

In Problems 15-26, write down the first five terms of each sequence.
$\left\{a_n\right\}=\left\{\frac{n}{n+2}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:28

Problem 17

In Problems 15-22, 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 fifty-first term?
$a_1=5 ; d=-3$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 17

In Problems 9-18, 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

Problem 17

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=\frac{1}{3} n(n+1)(n+2)$

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Problem 17

In Problems 17-28, expand each expression using the Binomial Theorem.
$(x+1)^5$

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00:58

Problem 18

In Problems 15-26, write down the first five terms of each sequence.
$\left\{b_n\right\}=\left\{\frac{2 n+1}{2 n}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:20

Problem 18

In Problems 15-22, 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 fifty-first term?
$a_1=6 ; d=-2$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:33

Problem 18

In Problems 9-18, 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

Problem 18

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$1 \cdot 2+3 \cdot 4+5 \cdot 6+\cdots+(2 n-1)(2 n)=\frac{1}{3} n(n+1)(4 n-1)$

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Problem 18

In Problems 17-28, expand each expression using the Binomial Theorem.
$(x-1)^5$

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00:59

Problem 19

In Problems 15-26, write down the first five terms of each sequence.
$\left\{c_n\right\}=\left\{(-1)^{n+1} n^2\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:17

Problem 19

In Problems 15-22, 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 fifty-first term?
$a_1=0 ; \quad d=\frac{1}{2}$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:44

Problem 19

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=2 ; \quad r=3$

Vishal Parmar
Vishal Parmar
Numerade Educator

Problem 19

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$n^2+n$ is divisible by 2 .

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Problem 19

In Problems 17-28, expand each expression using the Binomial Theorem.
$(x-2)^6$

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01:16

Problem 20

In Problems 15-26, write down the first five terms of each sequence.
$\left\{d_n\right\}=\left\{(-1)^{n-1}\left(\frac{n}{2 n-1}\right)\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:35

Problem 20

In Problems 15-22, 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 fifty-first term?
$a_1=1 ; d=-\frac{1}{3}$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:23

Problem 20

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=-2 ; \quad r=4$

Dale Sanford
Dale Sanford
Numerade Educator

Problem 20

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$n^3+2 n$ is divisible by 3 .

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Problem 20

In Problems 17-28, expand each expression using the Binomial Theorem.
$(x+3)^5$

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01:44

Problem 21

In Problems 15-26, write down the first five terms of each sequence.
$\left\{s_n\right\}=\left\{\frac{2^n}{3^n+1}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:15

Problem 21

In Problems 15-22, 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 fifty-first term?
$a_1=\sqrt{2} ; d=\sqrt{2}$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:51

Problem 21

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=5 ; \quad r=-1$

Vishal Parmar
Vishal Parmar
Numerade Educator

Problem 21

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$n^2-n+2$ is divisible by 2 .

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Problem 21

In Problems 17-28, expand each expression using the Binomial Theorem.
$(3 x+1)^4$

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00:44

Problem 22

In Problems 15-26, write down the first five terms of each sequence.
$\left\{s_n\right\}=\left\{\left(\frac{4}{3}\right)^n\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:17

Problem 22

In Problems 15-22, 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 fifty-first term?
$a_1=0 ; \quad d=\pi$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:50

Problem 22

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=6 ; \quad r=-2$

Vishal Parmar
Vishal Parmar
Numerade Educator

Problem 22

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$.
$n(n+1)(n+2)$ is divisible by 6 .

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Problem 22

In Problems 17-28, expand each expression using the Binomial Theorem.
$(2 x+3)^5$

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01:36

Problem 23

In Problems 15-26, write down the first five terms of each sequence.
$\left\{t_n\right\}=\left\{\frac{(-1)^n}{(n+1)(n+2)}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:32

Problem 23

In Problems 23-28, find the indicated term in each arithmetic sequence.
100 th term of $2,4,6, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:23

Problem 23

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=0 ; \quad r=\frac{1}{2}$

Dale Sanford
Dale Sanford
Numerade Educator
04:49

Problem 23

In Problems 23-27, prove each statement.
If $x>1$, then $x^n>1$.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 23

In Problems 17-28, expand each expression using the Binomial Theorem.
$\left(x^2+y^2\right)^5$

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00:48

Problem 24

In Problems 15-26, write down the first five terms of each sequence.
$\left\{a_n\right\}=\left\{\frac{3^n}{n}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:40

Problem 24

In Problems 23-28, find the indicated term in each arithmetic sequence.
80th term of $-1,1,3, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:08

Problem 24

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=1 ; \quad r=-\frac{1}{3}$

Dale Sanford
Dale Sanford
Numerade Educator
03:44

Problem 24

In Problems 23-27, prove each statement.
If $0<x<1$, then $0<x^n<1$.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 24

In Problems 17-28, expand each expression using the Binomial Theorem.
$\left(x^2-y^2\right)^6$

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00:32

Problem 25

In Problems 15-26, write down the first five terms of each sequence.
$\left\{b_n\right\}=\left\{\frac{n}{e^n}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:14

Problem 25

In Problems 23-28, find the indicated term in each arithmetic sequence.
90 th term of $1,-2,-5, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:55

Problem 25

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=\sqrt{2} ; \quad r=\sqrt{2}$

Vishal Parmar
Vishal Parmar
Numerade Educator
05:49

Problem 25

In Problems 23-27, prove each statement.
$a-b$ is a factor of $a^n-b^n$.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 25

In Problems 17-28, expand each expression using the Binomial Theorem.
$(\sqrt{x}+\sqrt{2})^6$

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00:47

Problem 26

In Problems 15-26, write down the first five terms of each sequence.
$\left\{c_n\right\}=\left\{\frac{n^2}{2^n}\right\}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:29

Problem 26

In Problems 23-28, find the indicated term in each arithmetic sequence.
80 th term of $5,0,-5, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:38

Problem 26

In Problems 19-26, find the fifth term and the nth term of the geometric sequence whose initial term $a_1$ and common ratio $r$ are given.
$a_1=0 ; \quad r=\frac{1}{\pi}$

Vishal Parmar
Vishal Parmar
Numerade Educator
05:49

Problem 26

In Problems 23-27, prove each statement.
$a+b$ is a factor of $a^{2 n+1}+b^{2 n+1}$.

Sirat Shah
Sirat Shah
Numerade Educator

Problem 26

In Problems 17-28, expand each expression using the Binomial Theorem.
$(\sqrt{x}-\sqrt{3})^4$

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00:37

Problem 27

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:29

Problem 27

In Problems 23-28, find the indicated term in each arithmetic sequence.
80 th term of $2, \frac{5}{2}, 3, \frac{7}{2}, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:49

Problem 27

In Problems 27-32, find the indicated term of each geometric sequence.
7 th term of $1, \frac{1}{2}, \frac{1}{4}, \ldots$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
05:46

Problem 27

In Problems 23-27, prove each statement.
$(1+a)^n \geq 1+n a$, for $a>0$

Sirat Shah
Sirat Shah
Numerade Educator

Problem 27

In Problems 17-28, expand each expression using the Binomial Theorem.
$(a x+b y)^5$

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00:32

Problem 28

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$\frac{1}{1 \cdot 2}, \frac{1}{2 \cdot 3}, \frac{1}{3 \cdot 4}, \frac{1}{4 \cdot 5}, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:52

Problem 28

In Problems 23-28, find the indicated term in each arithmetic sequence.
70 th term of $2 \sqrt{5}, 4 \sqrt{5}, 6 \sqrt{5}, \ldots$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:26

Problem 28

In Problems 27-32, find the indicated term of each geometric sequence.
8 th term of $1,3,9, \ldots$

Angela Guo
Angela Guo
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

Problem 28

In Problems 17-28, expand each expression using the Binomial Theorem.
$(a x-b y)^4$

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00:29

Problem 29

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:14

Problem 29

In Problems 29-36, 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.
8 th term is $8 ; 20$ th term is 44

Raushan Kumar
Raushan Kumar
Numerade Educator
01:07

Problem 29

In Problems 27-32, find the indicated term of each geometric sequence.
9 th term of $1,-1,1, \ldots$

Dale Sanford
Dale Sanford
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

Problem 29

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^6$ in the expansion of $(x+3)^{10}$

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00:43

Problem 30

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$\frac{2}{3}, \frac{4}{9}, \frac{8}{27}, \frac{16}{81}, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:50

Problem 30

In Problems 29-36, 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.
4 th term is $3 ; 20$ th term is 35

Raushan Kumar
Raushan Kumar
Numerade Educator
01:20

Problem 30

In Problems 27-32, find the indicated term of each geometric sequence.
10 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

Problem 30

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^3$ in the expansion of $(x-3)^{10}$

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00:29

Problem 31

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$1,-1,1,-1,1,-1, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:58

Problem 31

In Problems 29-36, 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.
9 th term is $-5 ; 15$ th term is 31

Raushan Kumar
Raushan Kumar
Numerade Educator
01:01

Problem 31

In Problems 27-32, find the indicated term of each geometric sequence.
8 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
$$
\begin{aligned}
& a+(a+d)+(a+2 d) \\
& +\cdots+[a+(n-1) d]=n a+d \frac{n(n-1)}{2}
\end{aligned}
$$

Sirat Shah
Sirat Shah
Numerade Educator

Problem 31

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^7$ in the expansion of $(2 x-1)^{12}$

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00:37

Problem 32

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:25

Problem 32

In Problems 29-36, 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.
8 th term is $4 ; 18$ th term is -96

Raushan Kumar
Raushan Kumar
Numerade Educator
00:44

Problem 32

In Problems 27-32, find the indicated term of each geometric sequence.
7 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 $\frac{1}{2} n(n-3)$.
[Hint: Begin by showing that the result is true when $n=4$ (Condition I).]

Sirat Shah
Sirat Shah
Numerade Educator

Problem 32

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^3$ in the expansion of $(2 x+1)^{12}$

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00:30

Problem 33

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$1,-2,3,-4,5,-6, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:02

Problem 33

In Problems 29-36, 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.
15 th term is $0 ; 40$ th term is -50

Raushan Kumar
Raushan Kumar
Numerade Educator
00:39

Problem 33

In Problems 33-40, 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

Geometry 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

Problem 33

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^7$ in the expansion of $(2 x+3)^9$

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00:32

Problem 34

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence $\left\{a_n\right\}$ suggested by the pattern.
$2,-4,6,-8,10, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:24

Problem 34

In Problems 29-36, 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 th term is $-2 ; 13$ th term is 30

Raushan Kumar
Raushan Kumar
Numerade Educator
00:39

Problem 34

In Problems 33-40, 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

Problem 34

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^2$ in the expansion of $(2 x-3)^9$

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01:17

Problem 35

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=2 ; a_n=3+a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:10

Problem 35

In Problems 29-36, 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.
14 th term is $-1 ; 18$ th term is -9

Raushan Kumar
Raushan Kumar
Numerade Educator
00:41

Problem 35

In Problems 33-40, find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$-3,1,-\frac{1}{3}, \frac{1}{9}, \ldots$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 35

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The fifth term in the expansion of $(x+3)^7$

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00:40

Problem 36

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=3 ; \quad a_n=4-a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:22

Problem 36

In Problems 29-36, 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.
12 th term is $4 ; 18$ th term is 28

Raushan Kumar
Raushan Kumar
Numerade Educator
00:35

Problem 36

In Problems 33-40, 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

Problem 36

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The third term in the expansion of $(x-3)^7$

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00:48

Problem 37

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=-2 ; \quad a_n=n+a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:38

Problem 37

In Problems 37-54, find each sum.
$1+3+5+\cdots+(2 n-1)$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:00

Problem 37

In Problems 33-40, 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

Problem 37

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The third term in the expansion of $(3 x-2)^9$

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00:47

Problem 38

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=1 ; \quad a_n=n-a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:49

Problem 38

In Problems 37-54, find each sum.
$2+4+6+\cdots+2 n$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:50

Problem 38

In Problems 33-40, 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

Problem 38

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The sixth term in the expansion of $(3 x+2)^8$

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00:53

Problem 39

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=5 ; \quad a_n=2 a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:39

Problem 39

In Problems 37-54, find each sum.
$7+12+17+\cdots+(2+5 n)$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:47

Problem 39

In Problems 33-40, find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$a_2=7 ; \quad a_4=1575$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 39

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^0$ in the expansion of $\left(x^2+\frac{1}{x}\right)^{12}$

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00:37

Problem 40

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=2 ; \quad a_n=-a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:49

Problem 40

In Problems 37-54, find each sum.
$-1+3+7+\cdots+(4 n-5)$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:50

Problem 40

In Problems 33-40, find the nth term $a_n$ of each geometric sequence. When given, $r$ is the common ratio.
$a_3=\frac{1}{3} ; \quad a_6=\frac{1}{81}$

Amit Srivastava
Amit Srivastava
Numerade Educator

Problem 40

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^0$ in the expansion of $\left(x-\frac{1}{x^2}\right)^9$

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00:54

Problem 41

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=3 ; \quad a_n=\frac{a_{n-1}}{n}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:24

Problem 41

In Problems 37-54, find each sum.
$2+4+6+\cdots+70$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:49

Problem 41

In Problems 41-46, find each sum.
$\frac{1}{4}+\frac{2}{4}+\frac{2^2}{4}+\frac{2^3}{4}+\cdots+\frac{2^{n-1}}{4}$

Raushan Kumar
Raushan Kumar
Numerade Educator

Problem 41

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^4$ in the expansion of $\left(x-\frac{2}{\sqrt{x}}\right)^{10}$

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00:48

Problem 42

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=-2 ; \quad a_n=n+3 a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:44

Problem 42

In Problems 37-54, find each sum.
$1+3+5+\cdots+59$

Raushan Kumar
Raushan Kumar
Numerade Educator
02:18

Problem 42

In Problems 41-46, find each sum.
$\frac{3}{9}+\frac{3^2}{9}+\frac{3^3}{9}+\cdots+\frac{3^n}{9}$

Raushan Kumar
Raushan Kumar
Numerade Educator

Problem 42

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term.
The coefficient of $x^2$ in the expansion of $\left(\sqrt{x}+\frac{3}{\sqrt{x}}\right)^8$

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00:49

Problem 43

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=1 ; \quad a_2=2 ; \quad a_n=a_{n-1} \cdot a_{n-2}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:18

Problem 43

In Problems 37-54, find each sum.
$5+9+13+\cdots+49$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:25

Problem 43

In Problems 41-46, find each sum.
$\sum_{k=1}^n\left(\frac{2}{3}\right)^k$

Anurag Kumar
Anurag Kumar
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
00:59

Problem 44

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=-1 ; \quad a_2=1 ; \quad a_n=a_{n-2}+n a_{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:24

Problem 44

In Problems 37-54, find each sum.
$2+5+8+\cdots+41$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:25

Problem 44

In Problems 41-46, find each sum.
$\sum_{k=1}^n 4 \cdot 3^{k-1}$

Anurag Kumar
Anurag Kumar
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
00:55

Problem 45

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=A ; \quad a_n=a_{n-1}+d$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:48

Problem 45

In Problems 37-54, find each sum.
$73+78+83+88+\cdots+558$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:49

Problem 45

In Problems 41-46, find each sum.
$-1-2-4-8-\cdots-\left(2^{n-1}\right)$

Raushan Kumar
Raushan Kumar
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
00:42

Problem 46

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=A ; \quad a_n=r a_{n-1}, \quad r \neq 0$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:00

Problem 46

In Problems 37-54, find each sum.
$7+1-5-11-\cdots-299$

Chris Wojturski
Chris Wojturski
Numerade Educator
01:49

Problem 46

In Problems 41-46, find each sum.
$2+\frac{6}{5}+\frac{18}{25}+\cdots+2\left(\frac{3}{5}\right)^{n-1}$

Raushan Kumar
Raushan Kumar
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
01:03

Problem 47

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=\sqrt{2} ; \quad a_n=\sqrt{2+a_{n-1}}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:31

Problem 47

In Problems 37-54, find each sum.
$4+4.5+5+5.5+\cdots+100$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:01

Problem 47

For Problems 47-52, 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
01:14

Problem 48

In Problems 35-48, a sequence is defined recursively. Write down the first five terms.
$a_1=\sqrt{2} ; \quad a_n=\sqrt{\frac{a_{n-1}}{2}}$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:25

Problem 48

In Problems 37-54, find each sum.
$8+8 \frac{1}{4}+8 \frac{1}{2}+8 \frac{3}{4}+9+\cdots+50$

Chris Wojturski
Chris Wojturski
Numerade Educator
00:48

Problem 48

For Problems 47-52, 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
00:30

Problem 49

In Problems 49-58, write out each sum.
$\sum_{k=1}^n(k+2)$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:38

Problem 49

In Problems 37-54, find each sum.
$\sum_{n=1}^{80}(2 n-5)$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:38

Problem 49

For Problems 47-52, 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

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

Grant Mansfield
Grant Mansfield
Numerade Educator
00:40

Problem 50

In Problems 49-58, write out each sum.
$\sum_{k=1}^n(2 k+1)$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:38

Problem 50

In Problems 37-54, find each sum.
$\sum_{n=1}^{90}(3-2 n)$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:38

Problem 50

For Problems 47-52, 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

Stirling's Formula 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
00:38

Problem 51

In Problems 49-58, write out each sum.
$\sum_{k=1}^n \frac{k^2}{2}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:38

Problem 51

In Problems 37-54, find each sum.
$\sum_{n=1}^{100}\left(6-\frac{1}{2} n\right)$

Raushan Kumar
Raushan Kumar
Numerade Educator
01:18

Problem 51

For Problems 47-52, 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
00:33

Problem 52

In Problems 49-58, write out each sum.
$\sum_{k=1}^n(k+1)^2$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:38

Problem 52

In Problems 37-54, find each sum.
$\sum_{n=1}^{80}\left(\frac{1}{3} n+\frac{1}{2}\right)$

Raushan Kumar
Raushan Kumar
Numerade Educator
00:45

Problem 52

For Problems 47-52, 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
01:26

Problem 53

In Problems 49-58, write out each sum.
$\sum_{k=0}^n \frac{1}{3^k}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:22

Problem 53

In Problems 37-54, find each sum.
The sum of the first 120 terms of the sequence $14,16,18$, $20, \ldots$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:39

Problem 53

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$1+\frac{1}{3}+\frac{1}{9}+\cdots$

Adrian Co
Adrian Co
Numerade Educator
01:27

Problem 54

In Problems 49-58, write out each sum.
$\sum_{k=0}^n\left(\frac{3}{2}\right)^k$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:33

Problem 54

In Problems 37-54, find each sum.
The sum of the first 46 terms of the sequence $2,-1,-4$, $-7, \ldots$.

Julie Silva
Julie Silva
Numerade Educator
01:13

Problem 54

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$2+\frac{4}{3}+\frac{8}{9}+\cdots$

Adrian Co
Adrian Co
Numerade Educator
01:25

Problem 55

In Problems 49-58, write out each sum.
$\sum_{k=0}^{n-1} \frac{1}{3^{k+1}}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:01

Problem 55

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
00:24

Problem 55

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$8+4+2+\cdots$

Adrian Co
Adrian Co
Numerade Educator
01:23

Problem 56

In Problems 49-58, write out each sum.
$\sum_{k=0}^{n-1}(2 k+1)$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:20

Problem 56

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
02:29

Problem 56

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$6+2+\frac{2}{3}+\cdots$

William Semus
William Semus
Numerade Educator
00:33

Problem 57

In Problems 49-58, write out each sum.
$\sum_{k=2}^n(-1)^k \ln k$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:26

Problem 57

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
00:24

Problem 57

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$2-\frac{1}{2}+\frac{1}{8}-\frac{1}{32}+\cdots$

Adrian Co
Adrian Co
Numerade Educator
02:05

Problem 58

In Problems 49-58, write out each sum.
$\sum_{k=3}^n(-1)^{k+1} 2^k$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:02

Problem 58

How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is -4 to obtain a sum of 702 ?

Julie Silva
Julie Silva
Numerade Educator
01:01

Problem 58

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$1-\frac{3}{4}+\frac{9}{16}-\frac{27}{64}+\cdots$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:19

Problem 59

In Problems 59-68, express each sum using summation notation.
$1+2+3+\cdots+20$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:40

Problem 59

Drury Lane Theater 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
00:24

Problem 59

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$8+12+18+27+\cdots$

Adrian Co
Adrian Co
Numerade Educator
00:19

Problem 60

In Problems 59-68, express each sum using summation notation.
$1^3+2^3+3^3+\cdots+8^3$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:30

Problem 60

Football Stadium 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?
(IMAGE CANT COPY)

Julie Silva
Julie Silva
Numerade Educator
00:37

Problem 60

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$9+12+16+\frac{64}{3}+\cdots$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:54

Problem 61

In Problems 59-68, express each sum using summation notation.
$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{13}{13+1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:10

Problem 61

Creating a Mosaic 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?
(IMAGE CANT COPY)

Julie Silva
Julie Silva
Numerade Educator
00:31

Problem 61

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 5\left(\frac{1}{4}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
00:39

Problem 62

In Problems 59-68, express each sum using summation notation.
$1+3+5+7+\cdots+[2(12)-1]$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:19

Problem 62

Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two less 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
00:31

Problem 62

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
01:13

Problem 63

In Problems 59-68, express each sum using summation notation.
$1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots+(-1)^6\left(\frac{1}{3^6}\right)$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:58

Problem 63

Cooling Air 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?
Source: National Aeronautics and Space Administration

Vishal Parmar
Vishal Parmar
Numerade Educator
00:31

Problem 63

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} \frac{1}{2} 3^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
00:53

Problem 64

In Problems 59-68, express each sum using summation notation.
$\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\cdots+(-1)^{12}\left(\frac{2}{3}\right)^{11}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:34

Problem 64

Citrus Ladders 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?

Vishal Parmar
Vishal Parmar
Numerade Educator
00:31

Problem 64

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
00:29

Problem 65

In Problems 59-68, express each sum using summation notation.
$3+\frac{3^2}{2}+\frac{3^3}{3}+\cdots+\frac{3^n}{n}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:37

Problem 65

Seats in an Amphitheater 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
00:31

Problem 65

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 6\left(-\frac{2}{3}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
00:29

Problem 66

In Problems 59-68, express each sum using summation notation.
$\frac{1}{e}+\frac{2}{e^2}+\frac{3}{e^3}+\cdots+\frac{n}{e^n}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:02

Problem 66

Stadium Construction 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?

Vishal Parmar
Vishal Parmar
Numerade Educator
00:31

Problem 66

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1}$

Adrian Co
Adrian Co
Numerade Educator
01:09

Problem 67

In Problems 59-68, express each sum using summation notation.
$a+(a+d)+(a+2 d)+\cdots+(a+n d)$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:04

Problem 67

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

Vishal Parmar
Vishal Parmar
Numerade Educator
00:31

Problem 67

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 3\left(\frac{2}{3}\right)^k$

Adrian Co
Adrian Co
Numerade Educator
00:30

Problem 68

In Problems 59-68, express each sum using summation notation.
$a+a r+a r^2+\cdots+a r^{n-1}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:01

Problem 68

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

Vishal Parmar
Vishal Parmar
Numerade Educator
00:31

Problem 68

In Problems 53-68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.
$\sum_{k=1}^{\infty} 2\left(\frac{3}{4}\right)^k$

Adrian Co
Adrian Co
Numerade Educator
00:23

Problem 69

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{40} 5$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:15

Problem 69

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

Joshua Eastwood
Joshua Eastwood
Numerade Educator

Problem 69

In Problems 69-82, 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.
$\{n+2\}$

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00:21

Problem 70

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{50} 8$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:01

Problem 70

In Problems 69-82, 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.
$\{2 n-5\}$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:37

Problem 71

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{40} k$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 71

In Problems 69-82, 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.
$\left\{4 n^2\right\}$

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00:51

Problem 72

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{24}(-k)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 72

In Problems 69-82, 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.
$\left\{5 n^2+1\right\}$

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01:15

Problem 73

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{20}(5 k+3)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 73

In Problems 69-82, 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.
$\left\{3-\frac{2}{3} n\right\}$

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01:34

Problem 74

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{26}(3 k-7)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 74

In Problems 69-82, 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.
$\left\{8-\frac{3}{4} n\right\}$

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01:24

Problem 75

In Problems 69-80, find the sum of each sequence.
$\sum_{k=1}^{16}\left(k^2+4\right)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 75

In Problems 69-82, 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.
$1,3,6,10, \ldots$

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01:14

Problem 76

In Problems 69-80, find the sum of each sequence.
$\sum_{k=0}^{14}\left(k^2-4\right)$

Anurag Kumar
Anurag Kumar
Numerade Educator
00:21

Problem 76

In Problems 69-82, 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.
$2,4,6,8, \ldots$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:00

Problem 77

In Problems 69-80, find the sum of each sequence.
$\sum_{k=10}^{60}(2 k)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 77

In Problems 69-82, 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.
$\left\{\left(\frac{2}{3}\right)^n\right\}$

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01:15

Problem 78

In Problems 69-80, find the sum of each sequence.
$\sum_{k=8}^{40}(-3 k)$

Anurag Kumar
Anurag Kumar
Numerade Educator

Problem 78

In Problems 69-82, 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.
$\left\{\left(\frac{5}{4}\right)^n\right\}$

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00:56

Problem 79

In Problems 69-80, find the sum of each sequence.
$\sum_{k=5}^{20} k^3$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:27

Problem 79

In Problems 69-82, 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.
$-1,2,-4,8, \ldots$.

Amit Srivastava
Amit Srivastava
Numerade Educator
01:09

Problem 80

In Problems 69-80, find the sum of each sequence.
$\sum_{k=4}^{24} k^3$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:25

Problem 80

In Problems 69-82, 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.
$1,1,2,3,5,8, \ldots$

Amit Srivastava
Amit Srivastava
Numerade Educator
00:58

Problem 81

Credit Card Debt John has a balance of $$\$ 3000$$ on his Discover card that charges $1 \%$ interest per month on any unpaid balance. 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 $B_1$.

Julie Silva
Julie Silva
Numerade Educator

Problem 81

In Problems 69-82, 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.
$\left\{3^{n / 2}\right\}$

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01:05

Problem 82

Trout Population A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing $3 \%$ 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 two months? That is, what is $p_2$ ?

Anurag Kumar
Anurag Kumar
Numerade Educator
01:17

Problem 82

In Problems 69-82, 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

Car Loans 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
01:48

Problem 84

Environmental Control 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

Growth of a Rabbit Colony 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?
(image cant copy)

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01:29

Problem 85

Salary Increases If you have been hired at an annual salary of $\$ 18,000$ and expect to receive annual increases of $5 \%$, what will your salary be when you begin your fifth year?

Vishal Parmar
Vishal Parmar
Numerade Educator
06:08

Problem 86

Fibonacci Sequence 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
02:01

Problem 86

Equipment Depreciation 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?

Amit Srivastava
Amit Srivastava
Numerade Educator
01:23

Problem 87

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

Julie Silva
Julie Silva
Numerade Educator
05:48

Problem 87

Pendulum Swings 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?

Amit Srivastava
Amit Srivastava
Numerade Educator
02:52

Problem 88

Fibonacci Sequence 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
02:20

Problem 88

Bouncing Balls 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.
(GRAPH CANT COPY)
(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 distance does the ball travel before it stops bouncing?

James Kiss
James Kiss
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
$$
f(x)=e^k \approx \sum_{k=0}^n \frac{x^k}{k !}
$$
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

Retirement 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 $f(x)=e^x \quad$ 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
01:31

Problem 90

Saving for a Home 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 36th deposit (3 years)?

Dale Sanford
Dale Sanford
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:
$$
a_1=0.4 \quad a_n=0.4+0.3 \cdot 2^{n-2}, n \geq 2
$$
(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 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 $30.07 \mathrm{AU}$ and $39.44 \mathrm{AU}$, 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.
where $n$ is the number of the planet from the sun.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:40

Problem 91

Tax Sheltered Annuity 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 80 th deposit ( 20 years) if the per annum rate of return is assumed to be $8 \%$ compounded quarterly?

Dale Sanford
Dale Sanford
Numerade Educator
01:29

Problem 92

Show that
$$
1+2+\cdots+(n-1)+n=\frac{n(n+1)}{2}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:44

Problem 92

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

Dale Sanford
Dale Sanford
Numerade Educator
02:01

Problem 93

Computing Square Roots 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.
$\sqrt{5}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:13

Problem 93

Sinking Fund 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?

Dale Sanford
Dale Sanford
Numerade Educator
01:37

Problem 94

Computing Square Roots 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.
$\sqrt{8}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:12

Problem 94

Sinking Fund For a child born in 1996, the cost of a 4-year college education at a public university is projected to be $$\$ 150,000$$. Assuming an $8 \%$ per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have $\$ 150,000$ in 18 years when the child begins college?

Dale Sanford
Dale Sanford
Numerade Educator
01:54

Problem 95

Computing Square Roots 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.
$\sqrt{21}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:50

Problem 95

Grains of Wheat on a Chess Board 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.)
(IMAGE CANT COPY)

James Kiss
James Kiss
Numerade Educator
02:06

Problem 96

Computing Square Roots 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.
$\sqrt{89}$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:43

Problem 96

Look at the figure. What fraction of the square is eventually shaded if the indicated shading process continues indefinitely?
(IMAGE CANT COPY)

Dale Sanford
Dale Sanford
Numerade Educator
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

Multiplier 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 that 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

Multiplier 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
$$
u_{n+1}+u_n=(n+1)^2 \text {. }
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:08

Problem 99

Stock Price 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:
$$
\text { Price }=P+P \frac{1+i}{1+r}+P\left(\frac{1+i}{1+r}\right)^2+P\left(\frac{1+i}{1+r}\right)^3+\cdots
$$
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
01:27

Problem 100

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

Julie Silva
Julie Silva
Numerade Educator
01:42

Problem 100

Stock Price 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
04:14

Problem 101

A Rich Man's Promise A rich man promises to give you $$\$ 1000$$ on September 1, 2010. 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 $1 c$ ? How much have you received when this happens?

Vishal Parmar
Vishal Parmar
Numerade Educator
03:17

Problem 102

Critical Thinking You are interviewing for a job and receive two offers:

A: $$\$ 20,000$$ to start, with guaranteed annual increases of $6 \%$ for the first 5 years
$B$ : $$\$ 22,000$$ 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 103

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
04:46

Problem 104

Critical Thinking 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?

Dale Sanford
Dale Sanford
Numerade Educator
03:39

Problem 105

Critical Thinking 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 would 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?

Julie Silva
Julie Silva
Numerade Educator
02:52

Problem 106

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

Amit Srivastava
Amit Srivastava
Numerade Educator
01:09

Problem 107

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 108

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 109

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

Amit Srivastava
Amit Srivastava
Numerade Educator