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Precalculus Metric Version

Ron Larson

Chapter 9

Sequences, Series, and Probability - all with Video Answers

Educators

Section 1

Sequences and Series

01:01

Problem 1

In Exercises 1-3, fill in the blanks.
When you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, the sequence is defined ________.

Anurag Kumar
Anurag Kumar
Numerade Educator
00:51

Problem 2

In Exercises 1-3, fill in the blanks.
For the $\operatorname{sum} \sum_{i=1}^n a_i, i$ is the ________ of summation, $n$ is the ________ limit of summation, and 1 is the ________ limit of summation.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:05

Problem 3

In Exercises 1-3, fill in the blanks.
The sum of the terms of a finite or infinite sequence is called a ________.

Jerry Zhang
Jerry Zhang
Numerade Educator
00:30

Problem 4

Which is the domain of an infinite sequence? a finite sequence?
(a) the first $n$ positive integers (b) the set of positive integers

AG
Ankit Gupta
Numerade Educator
00:18

Problem 5

Write $1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6$ in factorial notation.

AG
Ankit Gupta
Numerade Educator
00:16

Problem 6

What is another name for a finite series with 10 terms?

Vishal Parmar
Vishal Parmar
Numerade Educator

Problem 7

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=4 n-7$

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Problem 8

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=-2 n+8$

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Problem 9

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=(-1)^{n+1}+4$

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Problem 10

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=1-(-1)^n$

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Problem 11

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{2}{3}$

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Problem 12

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\left(\frac{1}{2}\right)^n$

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Problem 13

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=(-2)^n$

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Problem 14

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=6(-1)^{n+1}$

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Problem 15

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{1}{3} n^3$

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Problem 16

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{1}{n^2}$

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Problem 17

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{n}{n+2}$

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Problem 18

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{6 n}{3 n^2-1}$

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Problem 19

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=n(n-1)(n-2)$

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Problem 20

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=n\left(n^2-6\right)$

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Problem 21

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=(-1)^n\left(\frac{n}{n+1}\right)$

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Problem 22

In Exercises 7-22, write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$a_n=\frac{(-1)^{n+1}}{n^2+1}$

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00:37

Problem 23

In Exercises 23-26, find the missing term of the sequence.
$$
\begin{aligned}
& a_n=(-1)^n(3 n-2) \\
& a_{25}=
\end{aligned}
$$

Linh Vu
Linh Vu
Numerade Educator
00:33

Problem 24

In Exercises 23-26, find the missing term of the sequence.
$$
\begin{aligned}
& a_n=(-1)^{n-1}[n(n-1)] \\
& a_{16}=
\end{aligned}
$$

Linh Vu
Linh Vu
Numerade Educator
00:37

Problem 25

In Exercises 23-26, find the missing term of the sequence.
$\begin{aligned} & a_n=\frac{4 n}{2 n^2-3} \\ & a_{11}=\end{aligned}$

Linh Vu
Linh Vu
Numerade Educator
01:41

Problem 26

In Exercises 23-26, find the missing term of the sequence.
$\begin{aligned} & a_n=\frac{4 n^2-n+3}{n(n-1)(n+2)} \\ & a_{13}=\end{aligned}$

Aditya Sood
Aditya Sood
Numerade Educator
01:02

Problem 27

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=\frac{2}{3} n$

Linh Vu
Linh Vu
Numerade Educator
01:02

Problem 28

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=3 n+3(-1)^n$

Linh Vu
Linh Vu
Numerade Educator
00:44

Problem 29

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=16(-0.5)^{n-1}$

Linh Vu
Linh Vu
Numerade Educator
00:28

Problem 30

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=8(0.75)^{n-1}$

Linh Vu
Linh Vu
Numerade Educator
00:26

Problem 31

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=\frac{2 n}{n+1}$

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 32

In Exercises 27-32, use a graphing utility to graph the first 10 terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$a_n=\frac{3 n^2}{n^2+1}$

Linh Vu
Linh Vu
Numerade Educator

Problem 33

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$3,7,11,15,19, \ldots$.

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Problem 34

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$91,82,73,64,55, \ldots$.

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01:43

Problem 35

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$3,10,29,66,127$,

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator

Problem 36

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$0,3,8,15,24, \ldots$.

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00:19

Problem 37

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$1,-1,1,-1,1, \ldots$.

Linh Vu
Linh Vu
Numerade Educator

Problem 38

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$1,3,1,3,1, \ldots$

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00:47

Problem 39

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$-\frac{2}{3}, \frac{3}{4},-\frac{4}{5}, \frac{5}{6},-\frac{6}{7}, \ldots$.

Linh Vu
Linh Vu
Numerade Educator
00:31

Problem 40

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$\frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots$

Linh Vu
Linh Vu
Numerade Educator
04:30

Problem 41

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$\frac{2}{1}, \frac{3}{3}, \frac{4}{5}, \frac{5}{7}, \frac{6}{9}, \ldots$

Cindy Nadolny
Cindy Nadolny
Numerade Educator
01:07

Problem 42

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$\frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}, \ldots$

Linh Vu
Linh Vu
Numerade Educator
00:30

Problem 43

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}$,

Linh Vu
Linh Vu
Numerade Educator
01:43

Problem 44

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$2,3,7,25,121, \ldots$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:07

Problem 45

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$\frac{1}{1}, \frac{3}{1}, \frac{9}{2}, \frac{27}{6}, \frac{81}{24}$,

Linh Vu
Linh Vu
Numerade Educator
01:00

Problem 46

In Exercises 33-46, write an expression for the apparent $n$th term $a_n$ of the sequence. (Assume that $\boldsymbol{n}$ begins with 1.)
$\frac{2}{1}, \frac{6}{3}, \frac{24}{7}, \frac{120}{15}, \frac{720}{31}, \ldots$.

Jerry Zhang
Jerry Zhang
Numerade Educator
00:32

Problem 47

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_1=28, a_{k+1}=a_k-4$

Linh Vu
Linh Vu
Numerade Educator
01:19

Problem 48

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_1=3, a_{k+1}=2\left(a_k-1\right)$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
00:32

Problem 49

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_1=81, \quad a_{k+1}=\frac{1}{3} a_k$

Linh Vu
Linh Vu
Numerade Educator
01:07

Problem 50

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_1=14, a_{k+1}=(-2) a_k$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
01:07

Problem 51

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_0=1, \quad a_1=2, \quad a_k=a_{k-2}+\frac{1}{2} a_{k-1}$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
01:07

Problem 52

In Exercises 47-52, write the first five terms of the sequence defined recursively.
$a_0=-1, \quad a_1=1, \quad a_k=a_{k-2}+a_{k-1}$

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
02:22

Problem 53

In Exercises 53 and 54, use the Fibonacci sequence. (See Example 5.)
Write the first 12 terms of the Fibonacci sequence whose $n$th term is $a_n$ and the first 10 terms of the sequence given by $b_n=a_{n+1} / a_n, n \geq 1$.

Bob Bob
Bob Bob
Numerade Educator
01:29

Problem 54

In Exercises 53 and 54, use the Fibonacci sequence. (See Example 5.)
Using the definition for $b_n$ in Exercise 53 , show that $b_n$ can be defined recursively by $b_n=1+1 / b_{n-1}$.

Bob Bob
Bob Bob
Numerade Educator
01:25

Problem 55

In Exercises 55-58, write the first five terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 0 .)
$a_n=\frac{5}{n !}$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:16

Problem 56

In Exercises 55-58, write the first five terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 0 .)
$a_n=\frac{1}{(n+1) !}$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:16

Problem 57

In Exercises 55-58, write the first five terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 0 .)
$a_n=\frac{(-1)^n(n+3) !}{n !}$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:22

Problem 58

In Exercises 55-58, write the first five terms of the sequence. (Assume that $\boldsymbol{n}$ begins with 0 .)
$a_n=\frac{(-1)^{2 n+1}}{(2 n+1) !}$

Aditya Sood
Aditya Sood
Numerade Educator
00:22

Problem 59

In Exercises 59-62, simplify the factorial expression.
$\frac{4 !}{6 !}$

James Kiss
James Kiss
Numerade Educator
01:01

Problem 60

In Exercises 59-62, simplify the factorial expression.
$\frac{12 !}{4 ! \cdot 8 !}$

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 61

In Exercises 59-62, simplify the factorial expression.
$\frac{(n+1) !}{n !}$

Linh Vu
Linh Vu
Numerade Educator
00:44

Problem 62

In Exercises 59-62, simplify the factorial expression.
$\frac{(2 n-1) \text { ! }}{(2 n+1) !}$

Linh Vu
Linh Vu
Numerade Educator
01:51

Problem 63

In Exercises 63-70, find the sum.
$\sum_{i=1}^5(2 i-1)$

Aditya Sood
Aditya Sood
Numerade Educator
01:00

Problem 64

In Exercises 63-70, find the sum.
$\sum_{k=1}^4 10$

Bob Bob
Bob Bob
Numerade Educator
01:35

Problem 65

In Exercises 63-70, find the sum.
$\sum_{j=3}^5 \frac{1}{j^2-3}$

Aditya Sood
Aditya Sood
Numerade Educator
01:18

Problem 66

In Exercises 63-70, find the sum.
$\sum_{i=0}^4\left(3 i^2+5\right)$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:06

Problem 67

In Exercises 63-70, find the sum.
$\sum_{k=2}^5(k+1)^2(k-3)$

Aditya Sood
Aditya Sood
Numerade Educator
01:45

Problem 68

In Exercises 63-70, find the sum.
$\sum_{i=1}^4\left[(i-1)^2+(i+1)^3\right]$

Aditya Sood
Aditya Sood
Numerade Educator
01:35

Problem 69

In Exercises 63-70, find the sum.
$\sum_{i=1}^4 \frac{i !}{2^i}$

Aditya Sood
Aditya Sood
Numerade Educator
01:29

Problem 70

In Exercises 63-70, find the sum.
$\sum_{j=0}^5 \frac{(-1)^j}{j !}$

Aditya Sood
Aditya Sood
Numerade Educator
00:38

Problem 71

In Exercises 71-74, use a graphing utility to find the sum.
$\sum_{k=0}^4 \frac{(-1)^k}{k !}$

Linh Vu
Linh Vu
Numerade Educator
01:03

Problem 72

In Exercises 71-74, use a graphing utility to find the sum.
$\sum_{k=0}^4 \frac{(-1)^k}{k+1}$

Linh Vu
Linh Vu
Numerade Educator
01:18

Problem 73

In Exercises 71-74, use a graphing utility to find the sum.
$\sum_{n=0}^{25} \frac{1}{4^n}$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:18

Problem 74

In Exercises 71-74, use a graphing utility to find the sum.
$\sum_{n=0}^{10} \frac{n !}{2^n}$

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:17

Problem 75

In Exercises 75-84, use sigma notation to write the sum.
$\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(9)}$

Israel Hernandez
Israel Hernandez
Numerade Educator
02:21

Problem 76

In Exercises 75-84, use sigma notation to write the sum.
$\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\cdots+\frac{5}{1+15}$

Israel Hernandez
Israel Hernandez
Numerade Educator
00:49

Problem 77

In Exercises 75-84, use sigma notation to write the sum.
$\left[2\left(\frac{1}{8}\right)+3\right]+\left[2\left(\frac{2}{8}\right)+3\right]+\cdots+\left[2\left(\frac{8}{8}\right)+3\right]$

James Kiss
James Kiss
Numerade Educator
01:42

Problem 78

In Exercises 75-84, use sigma notation to write the sum.
$\left[1-\left(\frac{1}{6}\right)^2\right]+\left[1-\left(\frac{2}{6}\right)^2\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^2\right]$

James Kiss
James Kiss
Numerade Educator
01:43

Problem 79

In Exercises 75-84, use sigma notation to write the sum.
$3-9+27-81+243-729$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:42

Problem 80

In Exercises 75-84, use sigma notation to write the sum.
$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{128}$

James Kiss
James Kiss
Numerade Educator
02:20

Problem 81

In Exercises 75-84, use sigma notation to write the sum.
$\frac{1^2}{2}+\frac{2^2}{6}+\frac{3^2}{24}+\frac{4^2}{120}+\cdots+\frac{7^2}{40,320}$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:26

Problem 82

In Exercises 75-84, use sigma notation to write the sum.
$\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}+\cdots+\frac{1}{10 \cdot 12}$

Israel Hernandez
Israel Hernandez
Numerade Educator
01:30

Problem 83

In Exercises 75-84, use sigma notation to write the sum.
$\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}$

Linh Vu
Linh Vu
Numerade Educator
01:13

Problem 84

In Exercises 75-84, use sigma notation to write the sum.
$\frac{1}{2}+\frac{2}{4}+\frac{6}{8}+\frac{24}{16}+\frac{120}{32}+\frac{720}{64}$

Linh Vu
Linh Vu
Numerade Educator
02:29

Problem 85

In Exercises 85-88, find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$\sum_{i=1}^{\infty}\left(\frac{1}{2}\right)^i$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:09

Problem 86

In Exercises 85-88, find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$\sum_{i=1}^{\infty} 2\left(\frac{1}{3}\right)^i$

Linh Vu
Linh Vu
Numerade Educator
03:33

Problem 87

In Exercises 85-88, find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^n$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:45

Problem 88

In Exercises 85-88, find the (a) third, (b) fourth, and (c) fifth partial sums of the series.
$\sum_{n=1}^{\infty} 5\left(-\frac{1}{4}\right)^n$

Anurag Kumar
Anurag Kumar
Numerade Educator
00:42

Problem 89

In Exercises 89-92, find the sum of the infinite series.
$\sum_{i=1}^{\infty} \frac{6}{10^i}$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:35

Problem 90

In Exercises 89-92, find the sum of the infinite series.
$\sum_{k=1}^{\infty}\left(\frac{1}{10}\right)^k$

Aditya Sood
Aditya Sood
Numerade Educator
02:11

Problem 91

In Exercises 89-92, find the sum of the infinite series.
$\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^k$

Aditya Sood
Aditya Sood
Numerade Educator
00:42

Problem 92

In Exercises 89-92, find the sum of the infinite series.
$\sum_{i=1}^{\infty} \frac{2}{10^i}$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
06:47

Problem 93

An investor deposits $$\$ 10,000$$ in an account that earns $3.5 \%$ interest compounded quarterly. The balance in the account after $n$ quarters is given by
$$
A_n=10,000\left(1+\frac{0.035}{4}\right)^n, \quad n=1,2,3, \ldots
$$
(a) Write the first eight terms of the sequence.
(b) Find the balance in the account after 10 years by computing the 40th term of the sequence.
(c) Is the balance after 20 years twice the balance after 10 years? Explain.

Anurag Kumar
Anurag Kumar
Numerade Educator
02:37

Problem 94

The percent $p_n$ of U.S. adults who met federal physical activity guidelines from 2007 through 2018 can be approximated by
$$
\begin{aligned}
& p_n=0.0251 n^3-0.989 n^2+13.34 n-12.2, \\
& n=7,8, \ldots, 18
\end{aligned}
$$
where $n$ is the year, with $n=7$ corresponding to 2007. (Source:
National Center for Health Statistics)
(a) Write the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence.
(b) What can you conclude from the bar graph in part (a)?
(FIGURE CAN'T COPY)

Allison Knapp
Allison Knapp
Numerade Educator
01:56

Problem 95

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer.
$\sum_{i=1}^4\left(i^2+2 i\right)=\sum_{i=1}^4 i^2+2 \sum_{i=1}^4 i$

Aditya Sood
Aditya Sood
Numerade Educator
01:20

Problem 96

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer.
$\sum_{j=1}^4 2^j=\sum_{j=3}^6 2^{j-2}$

Jerry Zhang
Jerry Zhang
Numerade Educator
01:05

Problem 97

In Exercises 97-99, use the following definition of the arithmetic mean $\bar{x}$ of a set of $n$ measurements $x_1, x_2, x_3, \ldots, x_n$.
$\bar{x}=\frac{1}{n} \sum_{i=1}^n x_i$
Find the arithmetic mean of the six checking account balances $$\$327.15$$, $$\$785.69$$, $$\$433.04$$, $$\$265.38$$, $$\$ 604.12$$, and $$\$ 590.30$$. Use the statistical capabilities of a graphing utility to verify your result.

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
02:26

Problem 98

In Exercises 97-99, use the following definition of the arithmetic mean $\bar{x}$ of a set of $n$ measurements $x_1, x_2, x_3, \ldots, x_n$.
$\bar{x}=\frac{1}{n} \sum_{i=1}^n x_i$
Prove that $\sum_{i=1}^n\left(x_i-\bar{x}\right)=0$.

Yujie Wang
Yujie Wang
College of San Mateo
02:30

Problem 99

In Exercises 97-99, use the following definition of the arithmetic mean $\bar{x}$ of a set of $n$ measurements $x_1, x_2, x_3, \ldots, x_n$.
$\bar{x}=\frac{1}{n} \sum_{i=1}^n x_i$
Prove that
$$
\sum_{i=1}^n\left(x_i-\bar{x}\right)^2=\sum_{i=1}^n x_i^2-\frac{1}{n}\left(\sum_{i=1}^n x_i\right)^2 .
$$

Jacquelinne S. Mejia Sandoval
Jacquelinne S. Mejia Sandoval
Numerade Educator
00:35

Problem 100

The graph represents the first six terms of a sequence.
(a) Write the first six terms of the sequence.
(b) Write an expression for the apparent $n$th term $a_n$ of the sequence.
(c) Use sigma notation to represent the partial sum of the first 50 terms of the sequence.
(GRAPH CAN'T COPY)

AG
Ankit Gupta
Numerade Educator
02:53

Problem 101

In Exercises 101 and 102, describe the error in finding the sum.
$$
\begin{aligned}
\sum_{k=1}^4\left(3+2 k^2\right) & =\sum_{k=1}^4 3+\sum_{k=1}^4 2 k^2 \\
& =3+(2+8+18+32) \\
& =63
\end{aligned}
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
02:57

Problem 102

In Exercises 101 and 102, describe the error in finding the sum.
$$
\begin{aligned}
\sum_{n=0}^3(-1)^n n ! & =(-1)(1)+(1)(2)+(-1)(6) \\
& =-5
\end{aligned}
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
00:36

Problem 103

Write the first four terms of the sequence given by
$$
b_n=\frac{(-1)^{n+1}}{2 n+1} \text {. }
$$
Are the terms the same as the first four terms of $a_n$ given in Example 2? Explain.

Jacob Denson
Jacob Denson
Numerade Educator
06:07

Problem 104

A $3 \times 3 \times 3$ cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure.
(FIGURE CAN'T COPY)
(a) Determine how many unit cubes of the $3 \times 3 \times 3$ cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces.
(b) Repeat part (a) for a $4 \times 4 \times 4$ cube, a $5 \times 5 \times 5$ cube, and a $6 \times 6 \times 6$ cube.
(c) Write formulas you could use to repeat part (a) for an $n \times n \times n$ cube.

Anurag Kumar
Anurag Kumar
Numerade Educator
01:20

Problem 105

In Exercises 105-108, find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line.
$m=-\frac{1}{2},(0,3)$

Grace Muhihu
Grace Muhihu
Numerade Educator
01:40

Problem 106

In Exercises 105-108, find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line.
$m=-2,(-4,0)$

Karla Conrey
Karla Conrey
Numerade Educator
View

Problem 107

In Exercises 105-108, find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line.
$m=5,(1,1)$

Josie Rutledge
Josie Rutledge
Numerade Educator
01:23

Problem 108

In Exercises 105-108, find the slope-intercept form of the equation of the line that has the given slope and passes through the given point. Sketch the line.
$m=\frac{4}{3},(-3,5)$

Linh Vu
Linh Vu
Numerade Educator
00:51

Problem 109

In Exercises 109-112, find the vertices of the ellipse or hyperbola. Then sketch the conic.
$\frac{y^2}{9}-x^2=1$

James Kiss
James Kiss
Numerade Educator
00:46

Problem 110

In Exercises 109-112, find the vertices of the ellipse or hyperbola. Then sketch the conic.
$\frac{x^2}{25}+\frac{y^2}{49}=1$

James Kiss
James Kiss
Numerade Educator
00:37

Problem 111

In Exercises 109-112, find the vertices of the ellipse or hyperbola. Then sketch the conic.
$x^2+27 y^2=9$

Amy Jiang
Amy Jiang
Numerade Educator
00:51

Problem 112

In Exercises 109-112, find the vertices of the ellipse or hyperbola. Then sketch the conic.
$4 x^2-12 y^2=16$

James Kiss
James Kiss
Numerade Educator
00:49

Problem 113

In Exercises 113-116, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors $i$ and $j$.
$$
\begin{array}{ll}
\text { Initial Point } & \text { Terminal Point } \\
(4,1) & (6,-3)
\end{array}
$$

Suzanne W.
Suzanne W.
Numerade Educator
00:59

Problem 114

In Exercises 113-116, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors $i$ and $j$.
$$
\begin{array}{ll}
\text { Initial Point } & \text { Terminal Point } \\
(-2,0) & (-8,-1)
\end{array}
$$

Prashansha Kaushik
Prashansha Kaushik
Numerade Educator
01:59

Problem 115

In Exercises 113-116, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors $i$ and $j$.
$$
\begin{array}{ll}
\text { Initial Point } & \text { Terminal Point } \\
(5,-5) & (-4,0)
\end{array}
$$

Suzanne W.
Suzanne W.
Numerade Educator
00:49

Problem 116

In Exercises 113-116, the initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors $i$ and $j$.
$$
\begin{array}{ll}
\text { Initial Point } & \text { Terminal Point } \\
(-6,-9) & (-2,-7)
\end{array}
$$

Suzanne W.
Suzanne W.
Numerade Educator