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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 9

Infinite Series

Educators


Problem 1

In Exercises $1-10,$ write the first five terms of the sequence.
$$a_{n}=3^{n}$$

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Problem 2

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\frac{3^{n}}{n !}
$$

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Problem 3

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\left(-\frac{1}{4}\right)^{n}
$$

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Problem 4

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\left(-\frac{2}{3}\right)^{n}
$$

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Problem 5

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\sin \frac{n \pi}{2}
$$

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Problem 6

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\frac{2 n}{n+3}
$$

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Problem 7

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=\frac{(-1)^{n(n+1) / 2}}{n^{2}}
$$

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Problem 8

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)
$$

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Problem 9

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=5-\frac{1}{n}+\frac{1}{n^{2}}
$$

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Problem 10

In Exercises $1-10,$ write the first five terms of the sequence.
$$
a_{n}=10+\frac{2}{n}+\frac{6}{n^{2}}
$$

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Problem 11

In Exercises $11-14$ , write the first five terms of the recursively defined sequence.
$$
a_{1}=3, a_{k+1}=2\left(a_{k}-1\right)
$$

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Problem 12

In Exercises $11-14$ , write the first five terms of the recursively defined sequence.
$$
a_{1}=4, a_{k+1}=\left(\frac{k+1}{2}\right) a_{k}
$$

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Problem 13

In Exercises $11-14$ , write the first five terms of the recursively defined sequence.
$$
a_{1}=32, a_{k+1}=\frac{1}{2} a_{k}
$$

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Problem 14

In Exercises $11-14$ , write the first five terms of the recursively defined sequence.
$$
a_{1}=6, a_{k+1}=\frac{1}{3} a_{k}^{2}
$$

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Problem 15

In Exercises $15-18,$ match the sequence with its graph. IThe graphs are labeled (a), (b), (c), and (d).
$$
a_{n}=\frac{10}{n+1}
$$

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Problem 16

In Exercises $15-18,$ match the sequence with its graph. IThe graphs are labeled (a), (b), (c), and (d).
$$
a_{n}=\frac{10 n}{n+1}
$$

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Problem 17

In Exercises $15-18,$ match the sequence with its graph. IThe graphs are labeled (a), (b), (c), and (d).
$$
a_{n}=(-1)^{n}
$$

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Problem 18

In Exercises $15-18,$ match the sequence with its graph. IThe graphs are labeled (a), (b), (c), and (d).
$$
a_{n}=\frac{(-1)^{n}}{n}
$$

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Problem 19

In Exercises $19-22,$ match the sequence with the correct expression for its $n$ th term. [The $n$ th terms are labeled (a), (b), (c), and (d).]
$$
\begin{array}{ll}{\text { (a) } a_{n}=\frac{2}{3} n} & {\text { (b) } a_{n}=2-\frac{4}{n}} \\ {\text { (c) } a_{n}=16(-0.5)^{n-1}} & {\text { (d) } a_{n}=\frac{2 n}{n+1}}\end{array}
$$
$$
-2,0, \frac{2}{3}, 1, \ldots
$$

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Problem 20

In Exercises $19-22,$ match the sequence with the correct expression for its $n$ th term. [The $n$ th terms are labeled (a), (b), (c), and (d).]
$$
\begin{array}{ll}{\text { (a) } a_{n}=\frac{2}{3} n} & {\text { (b) } a_{n}=2-\frac{4}{n}} \\ {\text { (c) } a_{n}=16(-0.5)^{n-1}} & {\text { (d) } a_{n}=\frac{2 n}{n+1}}\end{array}
$$
$$
16,-8,4,-2, \dots
$$

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Problem 21

In Exercises $19-22,$ match the sequence with the correct expression for its $n$ th term. [The $n$ th terms are labeled (a), (b), (c), and (d).]
$$
\frac{2}{3}, \frac{4}{3}, 2, \frac{8}{3}, \dots
$$

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Problem 22

In Exercises $19-22,$ match the sequence with the correct expression for its $n$ th term. [The $n$ th terms are labeled (a), (b), (c), and (d).]
$$
1, \frac{4}{3}, \frac{3}{2}, \frac{8}{5}, \dots
$$

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Problem 23

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
2,5,8,11, \ldots
$$

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Problem 24

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
\frac{7}{2}, 4, \frac{9}{2}, 5, \dots
$$

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Problem 25

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
5,10,20,40, \dots
$$

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Problem 26

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \dots
$$

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Problem 27

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
3,-\frac{3}{2}, \frac{3}{4},-\frac{3}{8}, \dots
$$

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Problem 28

In Exercises $23-28$ , write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$
1,-\frac{3}{2}, \frac{9}{4},-\frac{27}{8}, \dots
$$

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Problem 29

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{11 !}{8 !}
$$

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Problem 30

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{25 !}{20 !}
$$

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Problem 31

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{(n+1) !}{n !}
$$

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Problem 32

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{(n+2) !}{n !}
$$

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Problem 33

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{(2 n-1) !}{(2 n+1) !}
$$

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Problem 34

In Exercises $29-34,$ simplify the ratio of factorials.
$$
\frac{(2 n+2) !}{(2 n) !}
$$

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Problem 35

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=\frac{5 n^{2}}{n^{2}+2}
$$

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Problem 36

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=5-\frac{1}{n^{2}}
$$

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Problem 37

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=\frac{2 n}{\sqrt{n^{2}+1}}
$$

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Problem 38

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=\frac{5 n}{\sqrt{n^{2}+4}}
$$

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Problem 39

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=\sin \frac{1}{n}
$$

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Problem 40

In Exercises $35-40,$ find the limit (if possible) of the sequence.
$$
a_{n}=\cos \frac{2}{n}
$$

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Problem 41

In Exercises $41-44$ , use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$
a_{n}=\frac{n+1}{n}
$$

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Problem 42

In Exercises $41-44$ , use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$
a_{n}=\frac{1}{n^{3 / 2}}
$$

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Problem 43

In Exercises $41-44$ , use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$
a_{n}=\cos \frac{n \pi}{2}
$$

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Problem 44

In Exercises $41-44$ , use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$
a_{n}=3-\frac{1}{2^{n}}
$$

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Problem 45

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=(0.3)^{n}-1
$$

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Problem 46

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=4-\frac{3}{n}
$$

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Problem 47

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{5}{n+2}
$$

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Problem 48

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{2}{n !}
$$

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Problem 49

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)
$$

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Problem 50

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=1+(-1)^{n}
$$

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Problem 51

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{3 n^{2}-n+4}{2 n^{2}+1}
$$

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Problem 52

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{\sqrt[3]{n}}{\sqrt[3]{n}+1}
$$

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Problem 53

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}{(2 n)^{n}}
$$

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Problem 54

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot(2 n-1)}{n !}
$$

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Problem 55

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{1+(-1)^{n}}{n}
$$

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Problem 56

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{1+(-1)^{n}}{n^{2}}
$$

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Problem 57

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{\ln \left(n^{3}\right)}{2 n}
$$

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Problem 58

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{\ln \sqrt{n}}{n}
$$

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Problem 59

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{3^{n}}{4^{n}}
$$

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Problem 60

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=(0.5)^{n}
$$

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Problem 61

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{(n+1) !}{n !}
$$

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Problem 62

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{(n-2) !}{n !}
$$

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Problem 63

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{n-1}{n}-\frac{n}{n-1}, \quad n \geq 2
$$

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Problem 64

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{n^{2}}{2 n+1}-\frac{n^{2}}{2 n-1}
$$

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Problem 65

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{n^{p}}{e^{n}}, \quad p>0
$$

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Problem 66

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=n \sin \frac{1}{n}
$$

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Problem 67

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=2^{1 / n}
$$

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Problem 68

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=-3^{-n}
$$

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Problem 69

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\left(1+\frac{k}{n}\right)^{n}
$$

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Problem 70

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}
$$

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Problem 71

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{\sin n}{n}
$$

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Problem 72

In Exercises $45-72$ , determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$
a_{n}=\frac{\cos \pi n}{n^{2}}
$$

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Problem 73

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1,4,7,10, \dots
$$

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Problem 74

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
3,7,11,15, \dots
$$

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Problem 75

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
-1,2,7,14,23, \dots
$$

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Problem 76

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \dots
$$

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Problem 77

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots
$$

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Problem 78

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
2,-1, \frac{1}{2},-\frac{1}{4}, \frac{1}{8}, \dots
$$

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Problem 79

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
2,1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \ldots
$$

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Problem 80

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1+\frac{1}{2}, 1+\frac{3}{4}, 1+\frac{7}{8}, 1+\frac{15}{16}, 1+\frac{31}{32}, \dots
$$

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Problem 81

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
\frac{1}{2 \cdot 3}, \frac{2}{3 \cdot 4}, \frac{3}{4 \cdot 5}, \frac{4}{5 \cdot 6}, \dots
$$

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Problem 82

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \dots
$$

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Problem 83

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1,-\frac{1}{1 \cdot 3}, \frac{1}{1 \cdot 3 \cdot 5},-\frac{1}{1 \cdot 3 \cdot 5 \cdot 7}, \cdots
$$

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Problem 84

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1, x, \frac{x^{2}}{2}, \frac{x^{3}}{6}, \frac{x^{4}}{24}, \frac{x^{5}}{120}, \dots
$$

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Problem 85

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
2,24,720,40,320,3,628,800, \ldots
$$

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Problem 86

In Exercises $73-86,$ write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$
1,6,120,5040,362,880, \dots
$$

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Problem 87

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=4-\frac{1}{n}
$$

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Problem 88

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\frac{3 n}{n+2}
$$

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Problem 89

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\frac{n}{2^{n+2}}
$$

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Problem 90

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=n e^{-n / 2}
$$

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Problem 91

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=(-1)^{n}\left(\frac{1}{n}\right)
$$

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Problem 92

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\left(-\frac{2}{3}\right)^{n}
$$

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Problem 93

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\left(\frac{2}{3}\right)^{n}
$$

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Problem 94

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\left(\frac{3}{2}\right)^{n}
$$

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Problem 95

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\sin \frac{n \pi}{6}
$$

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Problem 96

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\cos \frac{n \pi}{2}
$$

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Problem 97

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\frac{\cos n}{n}
$$

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Problem 98

In Exercises $87-98$ , determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$
a_{n}=\frac{\sin \sqrt{n}}{n}
$$

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Problem 99

In Exercises $99-102,$ (a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit.
$$
a_{n}=5+\frac{1}{n}
$$

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Problem 100

In Exercises $99-102,$ (a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit.
$$
a_{n}=4-\frac{3}{n}
$$

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Problem 101

In Exercises $99-102,$ (a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit.
$$
a_{n}=\frac{1}{3}\left(1-\frac{1}{3^{n}}\right)
$$

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Problem 102

In Exercises $99-102,$ (a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges and (b) use a graphing utility to graph the first 10 terms of the sequence and find its limit.
$$
a_{n}=4+\frac{1}{2^{n}}
$$

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Problem 103

Let $\left\{a_{n}\right\}$ be an increasing sequence such that $2 \leq a_{n} \leq 4 .$ Explain why $\left\{a_{n}\right\}$ has a limit. What can you conclude about the limit?

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Problem 104

Let $\left\{a_{n}\right\}$ be a monotonic sequence such that $a_{n} \leq 1 .$ Discuss the convergence of $\left\{a_{n}\right\} .$ If $\left\{a_{n}\right\}$ converges, what can you conclude about its limit?

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Problem 105

Compound Interest Consider the sequence $\left\{A_{n}\right\}$ whose $n$ th term is given by
$$
A_{n}=P\left(1+\frac{r}{12}\right)^{n}
$$
where $P$ is the principal, $A_{n}$ is the account balance after $n$ months, and $r$ is the interest rate compounded annually.
(a) Is $\left\{A_{n}\right\}$ a convergent sequence? Explain.
(b) Find the first 10 terms of the sequence if $P=\$ 10,000$ and $r=0.055 .$

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Problem 106

Compound Interest A deposit of $\$ 100$ is made at the beginning of each month in an account at an annual interest rate of 3$\%$ compounded monthly. The balance in the account after $n$ months is $A_{n}=100(401)\left(1.0025^{n}-1\right)$
(a) Compute the first six terms of the sequence $\left\{A_{n}\right\}$
(b) Find the balance in the account after 5 years by computing the 60 th term of the sequence.
(c) Find the balance in the account after 20 years by computing the 240th term of the sequence.

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Problem 107

Is it possible for a sequence to converge to two different numbers? If so, give an example. If not, explain why not.

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Problem 108

In your own words, define each of the following.
$\begin{array}{ll}{\text { (a) Sequence }} & {\text { (b) Convergence of a sequence }} \\ {\text { (c) Monotonic sequence }} & {\text { (d) Bounded sequence }}\end{array}$

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Problem 109

The graphs of two sequences are shown in the figures. Which graph represents the sequence with alternating signs? Explain. Graph cannot copy

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Problem 110

Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.)
(a) A monotonically increasing sequence that converges to 10
(b) A monotonically increasing bounded sequence that does not converge
(c) A sequence that converges to $\frac{3}{4}$
(d) An unbounded sequence that converges to 100

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Problem 111

Government Expenditures A government program that currently costs taxpayers $\$ 4.5$ billion per year is cut back by 20 percent per year.
(a) Write an expression for the amount budgeted for this program after $n$ years.
(b) Compute the budgets for the first 4 years.
(c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

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Problem 112

Inflation If the rate of inflation is 4$\frac{1}{2} \%$ per year and the average price of a car is currently $\$ 25,000$ , the average price after $n$ years is
$$P_{n}=\$ 25,000(1.045)^{n}$$
Compute the average prices for the next 5 years.

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Problem 113

Modeling Data The federal debts $a_{n}$ (in billions of dollars) of the United States from 2002 through 2006 are shown in the table, where $n$ represents the year, with $n=2$ corresponding to 2002 . (Source: U.S. Office of Management and Budget)
$$
\begin{array}{|c|c|c|c|c|c|}\hline n & {2} & {3} & {4} & {5} & {6} \\ \hline a_{n} & {6198.4} & {6760.0} & {7354.7} & {7905.3} & {8451.4} \\ \hline\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a model of the form
$$
a_{n}=b n^{2}+c n+d, \quad n=2,3,4,5,6
$$
for the data. Use the graphing utility to plot the points and graph the model.
(b) Use the model to predict the amount of the federal debt in the year 2012 .

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Problem 114

Modeling Data The per capita personal incomes $a_{n}$ in the United States from 1996 through 2006 are given below as ordered pairs of the form $\left(n, a_{n}\right),$ where $n$ represents the year, with $n=6$ corresponding to $1996 .$ (Source: U.S. Bureau of Economic Analysis)
$$
\begin{array}{l}{(6,24,176),(7,25,334),(8,26,880),(9,27,933)} \\ {(10,29,855),(11,30,572),(12,30,805),(13,31,469)} \\ {(14,33,102),(15,34,493),(16,36,313)}\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a model of the form
$$
a_{n}=b n+c, \quad n=6,7, \ldots, 16
$$
for the data. Graphically compare the points and the model.
(b) Use the model to predict per capital personal income in the year 2012 .

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Problem 115

Comparing Exponential and Factorial Growth Consider the sequence $a_{n}=10^{n} / n ! .$
(a) Find two consecutive terms that are equal in magnitude.
(b) Are the terms following those found in part (a) increasing or decreasing?
(c) In Section 8.7 , Exercises $73-78$ , it was shown that for "large" values of the independent variable an exponential function increases more rapidly than a polynomial function. From the result in part (b), what inference can you make about the rate of growth of an exponential function versus a factorial function for "large" integer values of $n ?$

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Problem 116

Compute the first six terms of the sequence
$$\left\{a_{n}\right\}=\left\{\left(1+\frac{1}{n}\right)^{n}\right\}$$
If the sequence converges, find its limit.

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Problem 117

Compute the first six terms of the sequence $\left\{a_{n}\right\}=\{\sqrt[n]{n}\} .$ If the sequence converges, find its limit.

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Problem 118

Prove that if $\left\{s_{n}\right\}$ converges to $L$ and $L>0$ , then there exists a number $N$ such that $s_{n}>0$ for $n>N .$

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Problem 119

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges to 3 and $\left\{b_{n}\right\}$ converges to $2,$ then $\left\{a_{n}+b_{n}\right\}$ converges to $5 .$

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Problem 120

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges, then $\lim _{n \rightarrow \infty}\left(a_{n}-a_{n+1}\right)=0$

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Problem 121

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $n>1,$ then $n !=n(n-1) !$

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Problem 122

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges, then $\left\{a_{n} / n\right\}$ converges to 0

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Problem 123

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges to 0 and $\left\{b_{n}\right\}$ is bounded, then $\left\{a_{n} b_{n}\right\}$ converges to $0 .$

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Problem 124

True or False? In Exercises $119-124$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ diverges and $\left\{b_{n}\right\}$ diverges, then $\left\{a_{n}+b_{n}\right\}$ diverges.

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Problem 125

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. $1170-\mathrm{ca} .1240$ ) encountered the sequence now bearing his name. The sequence is defined recursively as $a_{n+2}=a_{n}+a_{n+1},$ where $a_{1}=1$ and $a_{2}=1$
(a) Write the first 12 terms of the sequence.
(b) Write the first 10 terms of the sequence defined by
$$
b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1
$$
(c) Using the definition in part (b), show that
$$b_{n}=1+\frac{1}{b_{n-1}}$$
(d) The golden ratio $\rho$ can be defined by $\lim _{n \rightarrow \infty} b_{n}=\rho .$ Show that $\rho=1+1 / \rho$ and solve this equation for $\rho$

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Problem 126

Conjecture Let $x_{0}=1$ and consider the sequence $x_{n}$ given by the formula
$$
x_{n}=\frac{1}{2} x_{n-1}+\frac{1}{x_{n-1}}, \quad n=1,2, \ldots
$$
Use a graphing utility to compute the first 10 terms of the sequence and make a conjecture about the limit of the sequence.

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Problem 127

Consider the sequence $\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}, \ldots}$
(a) Compute the first five terms of this sequence.
(b) Write a recursion formula for $a_{n},$ for $n \geq 2$ .
(c) Find $\lim _{n \rightarrow \infty} a_{n}$

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Problem 128

Consider the sequence $\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots$
(a) Compute the first five terms of this sequence.
(b) Write a recursion formula for $a_{n},$ for $n \geq 2$ .
(c) Find $\lim _{n \rightarrow \infty} a_{n}$

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Problem 129

Consider the sequence $\left\{a_{n}\right\}$ where $a_{1}=\sqrt{k}, a_{n+1}=\sqrt{k+a_{n}}$ and $k>0$.
(a) Show that $\left\{a_{n}\right\}$ is increasing and bounded.
(b) Prove that $\lim _{n \rightarrow \infty} a_{n}$ exists.
(c) Find $\lim _{n \rightarrow \infty} a_{n}$ .

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Problem 130

Arithmetic-Geometric Mean Let $a_{0}>b_{0}>0 .$ Let $a_{1}$ be the arithmetic mean of $a_{0}$ and $b_{0}$ and let $b_{1}$ be the geometric mean of $a_{0}$ and $b_{0} .$
$$
\begin{array}{ll}{a_{1}=\frac{a_{0}+b_{0}}{2}} & {\text { Arithmetic mean }} \\ {b_{1}=\sqrt{a_{0} b_{0}}} & {\text { Geometric mean }}\end{array}
$$
Now define the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ as follows.
$$
a_{n}=\frac{a_{n-1}+b_{n-1}}{2} \quad b_{n}=\sqrt{a_{n-1} b_{n-1}}
$$
(a) Let $a_{0}=10$ and $b_{0}=3 .$ Write out the first five terms of $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ . Compare the terms of $\left\{b_{n}\right\} .$ Compare $a_{n}$ and $b_{n} .$ What do you notice?
(c) Explain why $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are both convergent.
(d) Show that $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}$

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Problem 131

(a) Let $f(x)=\sin x$ and $a_{n}=n \sin 1 / n .$ Show that $\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0)=1 .$
(b) Let $f(x)$ be differentiable on the interval $[0,1]$ and $f(0)=0 .$ Consider the sequence $\left\{a_{n}\right\},$ where $a_{n}=n f(1 / n) .$ Show that $\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0)$

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Problem 132

Consider the sequence $\left\{a_{n}\right\}=\left\{n r^{n}\right\} .$ Decide whether $\left\{a_{n}\right\}$ converges for each value of $r .$
(a) $r=\frac{1}{2} \quad$ (b) $r=1 \quad$ (c) $r=\frac{3}{2}$
(d) For what values of $r$ does the sequence $\left\{n r^{n}\right\}$ converge?

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Problem 133

(a) Show that $\int_{1}^{n} \ln x d x<\ln (n !)$ for $n \geq 2$
Graph cannot copy
(b) Draw a graph similar to the one above that shows
$\ln (n !)<\int_{1}^{n+1} \ln x d x .$
(c) Use the results of parts (a) and (b) to show that
$$
\frac{n^{n}}{e^{n-1}}<n !<\frac{(n+1)^{n+1}}{e^{n}}, \text { for } n>1
$$
(d) Use the Squeeze Theorem for Sequences and the result of part (c) to show that $\lim _{n \rightarrow \infty}(\sqrt[n]{n !} / n)=1 / e$
(e) Test the result of part (d) for $n=20,50,$ and $100 .$

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Problem 134

Consider the sequence $\left\{a_{n}\right\}=\left\{\frac{1}{n} \sum_{k=1}^{n} \frac{1}{1+(k / n)}\right\}$
(a) Write the first five terms of $\left\{a_{n}\right\}$
(b) Show that $\lim _{n \rightarrow \infty} a_{n}=\ln 2$ by interpreting $a_{n}$ as a Riemann sum of a definite integral.

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Problem 135

Prove, using the definition of the limit of a sequence, that $\lim _{n \rightarrow \infty} \frac{1}{n^{3}}=0$

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Problem 136

Prove, using the definition of the limit of a sequence, that $\lim _{n \rightarrow \infty} r^{n}=0$ for $-1< r <1$

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Problem 137

Find a divergent sequence $\left\{a_{n}\right\}$ such that $\left\{a_{2 n}\right\}$ converges.

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Problem 138

Show that the converse of Theorem 9.1 is not true. [Hint: Find a function $f(x)$ such that $f(n)=a_{n}$ converges but $\lim _{x \rightarrow \infty} f(x)$ does not exist. $]$

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Problem 139

Prove Theorem 9.5 for a nonincreasing sequence.

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Problem 140

Let $\left\{x_{n}\right\}, n \geq 0,$ be a sequence of nonzero real numbers such that $x_{n}^{2}-x_{n-1} x_{n+1}=1$ for $n=1,2,3, \ldots$ Prove that there exists a real number $a$ such that $x_{n+1}=a x_{n}-x_{n-1},$ for all $n \geq 1$

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Problem 141

Let $T_{0}=2, T_{1}=3, T_{2}=6,$ and, for $n \geq 3$ $T_{n}=(n+4) T_{n-1}-4 n T_{n-2}+(4 n-8) T_{n-3}$
The first 10 terms of the sequence are $2,3,6,14,40,152,784,5168,40,576,363,392$ Find, with proof, a formula for $T_{n}$ of the form $T_{n}=A_{n}+B_{n},$ where $\left\{A_{n}\right\}$ and $\left\{B_{n}\right\}$ are well-known sequences.

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