Problem 1

$\mathrm{An}$ _____ _____ is a function whose domain is the set of positive integers.

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Problem 2

A sequence is a _____ sequence when the domain of the function consists only of the first $n$ positive integers.

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Problem 3

If 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, then the sequence is said to be defined _____.

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Problem 4

If $n$ is a positive integer, then $n$ _____ is defined as $n !=1 \cdot 2 \cdot 3 \cdot 4 \cdots(n-1) \cdot n$

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Problem 5

For the sum $$\sum_{i=1}^{n} a_{i}, $$ $i$ is called the _____ of summation, $n$ is the _____ limit of summation, and 1 is the _____ limit of summation.

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Problem 7

Writing the Terms of a Sequence 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

Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=2-\frac{1}{3^{n}}

$$

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Problem 9

Writing the Terms of a Sequence 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 10

Writing the Terms of a Sequence 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 11

Writing the Terms of a Sequence 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 12

Writing the Terms of a Sequence 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 13

Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{1+(-1)^{n}}{n}

$$

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Problem 14

Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{(-1)^{n}}{n^{2}}

$$

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Problem 15

Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{2^{n}}{3^{n}}

$$

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Problem 16

Writing the Terms of a Sequence In Exercises

$7-22,$ write the first five terms of the sequence. (Assume

that $n$ begins with $1 .$ )

$$

a_{n}=\frac{1}{n^{3 / 2}}

$$

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Problem 17

Writing the Terms of a Sequence 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 18

Writing the Terms of a Sequence 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 19

Writing the Terms of a Sequence 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

Writing the Terms of a Sequence 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

Writing the Terms of a Sequence 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

Writing the Terms of a Sequence 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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Problem 23

Finding a Term of a Sequence In Exercises $23-26$ ,

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=(-1)^{n}(3 n-2)} \\ {a_{25}=}\end{array}

$$

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Problem 24

Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=(-1)^{n-1}[n(n-1)]} \\ {a_{16}=}\end{array}

$$

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Problem 25

Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=\frac{4 n}{2 n^{2}-3}} \\ {a_{11}=}\end{array}

$$

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Problem 26

Finding a Term of a Sequence In Exercises $23-26$

find the indicated term of the sequence.

$$

\begin{array}{l}{a_{n}=\frac{4 n^{2}-n+3}{n(n-1)(n+2)}} \\ {a_{13}=}\end{array}

$$

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Problem 27

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{2}{3} n

$$

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Problem 28

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 .$ .

$$

a_{n}=2-\frac{4}{n}

$$

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Problem 29

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=16(-0.5)^{n-1}

$$

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Problem 30

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=8(0.75)^{n-1}

$$

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Problem 31

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{2 n}{n+1}

$$

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Problem 32

Graphing the Terms of a Sequence In Exercises

$27-32,$ use a graphing utility to graph the first 10 terms

of the sequence. (Assume that $n$ begins with $1 . )$

$$

a_{n}=\frac{3 n^{2}}{n^{2}+1}

$$

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Problem 33

Matching a Sequence with a Graph In Exercises

$33-36,$ match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{8}{n+1}

$$

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Problem 34

Matching a Sequence with a Graph In Exercises

$33-36$ , match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{8 n}{n+1}

$$

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Problem 35

Matching a Sequence with a Graph In Exercises

$33-36$ , match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), ( b), (c), and (d).]

$$

a_{n}=4(0.5)^{n-1}

$$

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Problem 36

Matching a Sequence with a Graph In Exercises

$33-36,$ match the sequence with the graph of its first 10

terms. [The graphs are labeled (a), (b), (c), and (d).]

$$

a_{n}=\frac{4^{n}}{n !}

$$

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Problem 37

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

3,7,11,15,19, \ldots

$$

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Problem 38

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

0,3,8,15,24, \dots

$$

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Problem 39

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

-\frac{2}{3}, \frac{3}{4},-\frac{4}{5}, \frac{5}{6},-\frac{6}{7}, \dots

$$

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Problem 40

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots

$$

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Problem 41

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{2}{1}, \frac{3}{3}, \frac{4}{5}, \frac{5}{7}, \frac{6}{9}, \ldots

$$

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Problem 42

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

\frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}, \dots

$$

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Problem 43

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \dots

$$

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Problem 44

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \dots

$$

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Problem 45

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,-1,1,-1,1, \ldots

$$

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Problem 46

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,3,1,3,1, \dots

$$

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Problem 47

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1,3, \frac{3^{2}}{2}, \frac{3^{3}}{6}, \frac{3^{4}}{24}, \frac{3^{5}}{120}, \dots

$$

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Problem 48

Finding the $n$ th Term of a Sequence In Exercises

$37-48,$ write an expression for the apparent $n$ th term

$\left(a_{n}\right)$ of the sequence. (Assume that $n$ begins with $1 . )$

$$

1+\frac{1}{2}, 1+\frac{3}{4}, 1+\frac{7}{8}, 1+\frac{15}{16}, 1+\frac{31}{32}, \ldots

$$

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Problem 49

Writing the Terms of a Recursive Sequence In

Exercises $49-52,$ write the first five terms of the sequence

defined recursivelv.

$$

a_{1}=28, \quad a_{k+1}=a_{k}-4

$$

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Problem 50

Writing the Terms of a Recursive Sequence In

Exercises $49-52$ , write the first five terms of the sequence

defined recursively.

$$

a_{1}=3, \quad a_{k+1}=2\left(a_{k}-1\right)

$$

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Problem 51

Writing the Terms of a Recursive Sequence In

Exercises $49-52$ , write the first five terms of the sequence

defined recursively.

$$

a_{0}=1, a_{1}=2, \quad a_{k}=a_{k-2}+\frac{1}{2} a_{k-1}

$$

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Problem 52

Writing the Terms of a Recursive Sequence In

Exercises $49-52,$ write the first five terms of the sequence

defined recursively.

$$

a_{0}=-1, a_{1}=1, \quad a_{k}=a_{k-2}+a_{k-1}

$$

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Problem 53

Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=6, \quad a_{k+1}=a_{k}+2

$$

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Problem 54

Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=25, \quad a_{k+1}=a_{k}-5

$$

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Problem 55

Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=81, \quad a_{k+1}=\frac{1}{3} a_{k}

$$

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Problem 56

Writing the $n$ th Term of a Recursive Sequence

In Exercises $53-56$ , write the first five terms of the

sequence defined recursively. Use the pattern to write the

$n$ th term of the sequence as a function of $n .$

$$

a_{1}=14, \quad a_{k+1}=(-2) a_{k}

$$

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Problem 57

Fibonacci Sequence In Exercises 57 and 58 , use the

Fibonacci sequence. (See Example 5.)

Write the first 12 terms of the Fibonacci sequence $a_{n}$

and the first 10 terms of the sequence given by

$$

b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1

$$

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Problem 58

Fibonacci Sequence In Exercises 57 and 58 , use the

Fibonacci sequence. (See Example $5 . )$

Using the definition for $b_{n}$ in Exercise $57,$ show that $b_{n}$

can be defined recursively by

$$

b_{n}=1+\frac{1}{b_{n-1}}

$$

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Problem 59

Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{5}{n !}

$$

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Problem 60

Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{n !}{2 n+1}

$$

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Problem 61

Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{1}{(n+1) !}

$$

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Problem 62

Writing the Terms of a Sequence Involving

Factorials In Exercises $59-62,$ write the first five terms

of the sequence. (Assume that $n$ begins with $0 . )$

$$

a_{n}=\frac{(-1)^{2 n+1}}{(2 n+1) !}

$$

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Problem 63

Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{4 !}{6 !}

$$

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Problem 64

Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{12 !}{4 ! \cdot 8 !}

$$

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Problem 65

Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{(n+1) !}{n !}

$$

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Problem 66

Simplifying a Factorial Expression In Exercises

$63-66,$ simplify the factorial expression.

$$

\frac{(2 n-1) !}{(2 n+1) !}

$$

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Problem 67

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{5}(2 i+1)

$$

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Problem 68

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{j=3}^{5} \frac{1}{j^{2}-3}

$$

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Problem 70

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=0}^{4} i^{2}

$$

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Problem 71

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{k=2}^{5}(k+1)^{2}(k-3)

$$

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Problem 72

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{4}\left[(i-1)^{2}+(i+1)^{3}\right]

$$

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Problem 73

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{i=1}^{4} 2^{i}

$$

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Problem 74

Finding a Sum In Exercises $67-74,$ find the sum.

$$

\sum_{j=0}^{4}(-2)^{j}

$$

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Problem 75

Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{n=0}^{5} \frac{1}{2 n+1}

$$

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Problem 76

Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{k=0}^{4} \frac{(-1)^{k}}{k+1}

$$

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Problem 77

Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}

$$

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Problem 78

Finding a Sum In Exercises $75-78$ , use a graphing

utility to find the sum.

$$

\sum_{n=0}^{25} \frac{1}{4^{n}}

$$

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Problem 79

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\dots+\frac{1}{3(9)}

$$

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Problem 80

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{5}{1+1}+\frac{5}{1+2}+\frac{5}{1+3}+\dots+\frac{5}{1+15}

$$

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Problem 81

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , 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]

$$

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Problem 82

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , 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]

$$

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Problem 83

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

3-9+27-81+243-729

$$

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Problem 84

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\dots-\frac{1}{128}

$$

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Problem 85

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\ldots-\frac{1}{20^{2}}

$$

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Problem 86

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{1 \cdot 3}+\frac{1}{2 \cdot 4}+\frac{1}{3 \cdot 5}+\dots+\frac{1}{10 \cdot 12}

$$

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Problem 87

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , use sigma notation to write the sum.

$$

\frac{1}{4}+\frac{3}{8}+\frac{7}{16}+\frac{15}{32}+\frac{31}{64}

$$

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Problem 88

Using Sigma Notation to Write a Sum In

Exercises $79-88$ , 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}

$$

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Problem 89

Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$

\sum_{i=1}^{\infty} 5\left(\frac{1}{2}\right)^{i}

$$ Fourth partial sum

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Problem 90

Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{i=1}^{\infty} 2\left(\frac{1}{3}\right)^{i}$$ Fifth partial sum

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Problem 91

Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{n=1}^{\infty} 4\left(-\frac{1}{2}\right)^{n}$$ Third partial sum

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Problem 92

Finding a Partial Sum of a Series In Exercises

$89-92,$ find the indicated partial sum of the series.

$$\sum_{n=1}^{\infty} 8\left(-\frac{1}{4}\right)^{n}$$ Fourth partial sum

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Problem 93

Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{i=1}^{\infty} \frac{6}{10^{i}}

$$

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Problem 94

Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{k=1}^{\infty}\left(\frac{1}{10}\right)^{k}

$$

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Problem 95

Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{k=1}^{\infty} 7\left(\frac{1}{10}\right)^{k}

$$

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Problem 96

Finding the Sum of an Infinite Series

Exercises $93-96$ , find the sum of the infinite series.

$$

\sum_{i=1}^{\infty} \frac{2}{10^{i}}

$$

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Problem 97

Compound Interest 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 40 th term of the sequence.

(c) Is the balance after 20 years twice the balance after

10 years? Explain.

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Problem 98

The numbers $a_{n}$ (in thousands) of AIDS cases reported

from 2003 through 2010 can be approximated by $a_{n}=-0.0126 n^{3}+0.391 n^{2}-4.21 n+48.5$

$n=3,4, \ldots, 10$

where $n$ is the

year, with $n=3$

corresponding to

$2003 .$ (Source:

U.S. Centers for

Disease Control

and Prevention)

(a) Write the terms of this finite sequence. Use a

graphing utility to construct a bar graph that

represents the sequence.

(b) What does the graph in part (a) say about reported

cases of AIDS?

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Problem 99

True or False? In Exercises 99 and $100,$ 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

$$

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Problem 100

True or False? In Exercises 99 and $100,$ determine

whether the statement is true or false. Justify your answer.

True or False? In Exercises 99 and $100,$ determine

whether the statement is true or false. Justify your answer.

$$

\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}

$$

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Problem 101

Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Find the arithmetic mean of the six checking account

balances $\$ 327.15, \quad \$ 785.69, \quad \$ 433.04, \quad \$ 265.38$

$\$ 604.12,$ and $\$ 590.30 .$ Use the statistical capabilities

of a graphing utility to verify your result.

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Problem 102

Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Proof Prove that $$\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)=0$$

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Problem 103

Arithmetic Mean In Exercises $101-103,$ use the

following definition of the arithmetic mean $\overline{x}$ of a set of $n$

measurements $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$

$$

\overline{x}=\frac{1}{n} \sum_{i=1}^{n} x_{i}

$$

Proof Prove that $$\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}=\sum_{i=1}^{n} x_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right)^{2}$$

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Problem 104

HOW DO YOU SEE IT? The graph

represents the first 10 terms of a sequence.

Complete each expression for the apparent $n$ th

term $a_{n}$ of the sequence. Which expressions

are appropriate to represent the cost $a_{n}$ to buy

$n \mathrm{MP} 3$ songs at a cost of $\$ 1$ per song?

Explain.

$$

a_{n}=1

$$

$$

a_{n}=\frac{!}{(n-1) !}

$$

$$

a_{n}=\sum_{k=1}^{n}

$$

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Problem 105

Finding the Terms of a Sequence In Exercises 105

and $106,$ find the first five terms of the sequence.

$$

a_{n}=\frac{x^{n}}{n !}

$$

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Problem 106

Finding the Terms of a Sequence In Exercises 105

and $106,$ find the first five terms of the sequence.

$$

a_{n}=\frac{(-1)^{n} x^{2 n+1}}{2 n+1}

$$

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Problem 107

Cube $A 3 \times 3 \times 3$ 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.

(a) Complete the table to 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 $66 \times 6 \times 6$ cube.

(c) What type of pattern do you observe?

(d) Write formulas you could use to repeat part (a) for

$\quad$ an $n \times n \times n$ cube.

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