Intermediate Algebra

Elayn Martin-Gay

Chapter 11

Sequences, Series, and the Binomial Theorem - all with Video Answers

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Section 1

Sequences

Problem 1

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=n+4
$$

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Problem 2

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=5-n
$$

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Problem 3

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=(-1)^{n}
$$

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Problem 4

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=(-2)^{n}
$$

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Problem 5

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=\frac{1}{n+3}
$$

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Problem 6

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=\frac{1}{7-n}
$$

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Problem 7

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=2 n
$$

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Problem 8

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=-6 n
$$

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Problem 9

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=-n^{2}
$$

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Problem 10

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=n^{2}+2
$$

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Problem 11

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=2^{n}
$$

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Problem 12

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=3^{n-2}
$$

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Problem 13

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=2 n+5
$$

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Problem 14

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=1-3 n
$$

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Problem 15

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=(-1)^{n} n^{2}
$$

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Problem 16

Write the first five terms of each sequence, whose general term is given. See Example 1
$$
a_{n}=(-1)^{n+1}(n-1)
$$

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Problem 17

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=3 n^{2} ; a_{5}
$$

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Problem 18

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=-n^{2} ; a_{15}
$$

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Problem 19

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=6 n-2 ; a_{20}
$$

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Problem 20

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=100-7 n ; a_{50}
$$

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Problem 21

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{n+3}{n}, a_{15}
$$

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Problem 22

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{n}{n+4}, a_{24}
$$

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Problem 23

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=(-3)^{n} ; a_{6}
$$

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Problem 24

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=5^{n+1} ; a_{3}
$$

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Problem 25

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{n-2}{n+1} ; a_{6}
$$

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Problem 26

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{n+3}{n+4} ; a_{8}
$$

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Problem 27

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{(-1)^{n}}{n}, a_{8}
$$

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Problem 28

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{(-1)^{n}}{2 n} ; a_{100}
$$

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Problem 29

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=-n^{2}+5 ; a_{10}
$$

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Problem 30

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=8-n^{2} ; a_{20}
$$

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Problem 31

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{(-1)^{n}}{n+6} ; a_{19}
$$

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Problem 32

Find the indicated term for each sequence, whose general term is given. See Example 2.
$$
a_{n}=\frac{n-4}{(-2)^{n}}, a_{6}
$$

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Problem 33

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
3,7,11,15
$$

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Problem 34

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
2,7,12,17
$$

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Problem 35

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
-2,-4,-8,-16
$$

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Problem 36

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
-4,16,-64,256
$$

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Problem 37

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}
$$

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Problem 38

Find a general term $a_{n}$ for each sequence, whose first four terms are given. See Example 3.
$$
\frac{2}{5}, \frac{2}{25}, \frac{2}{125}, \frac{2}{625}
$$

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Problem 39

Solve. See Example 4
The distance, in feet, that a Thermos dropped from a cliff falls in each consecutive second is modeled by a sequence whose general term is $a_{n}=32 n-16,$ where $n$ is the number of seconds. Find the distance the Thermos falls in the second, third, and fourth seconds.

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Problem 40

Solve. See Example 4
The population size of a culture of bacteria triples every hour such that its size is modeled by the sequence $a_{n}=50(3)^{n-1}$ where $n$ is the number of the hour just beginning. Find the size of the culture at the beginning of the fourth hour and the size of the culture at the beginning of the first hour.

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Problem 41

Solve. See Example 4
Mrs. Laser agrees to give her son Mark an allowance of $\$ 0.10$ on the first day of his 14 -day vacation, $\$ 0.20$ on the second day, $\$ 0.40$ on the third day, and so on. Write an equation of a sequence whose terms correspond to Mark's allowance. Find the allowance Mark will receive on the last day of his vacation.

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Problem 42

Solve. See Example 4
A small theater has 10 rows with 12 seats in the first row, 15 seats in the second row, 18 seats in the third row, and so on. Write an equation of a sequence whose terms correspond to the seats in each row. Find the number of seats in the eighth row.

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Problem 43

Solve. See Example 4
The number of cases of a new infectious disease is doubling every year such that the number of cases is modeled by a sequence whose general term is $a_{n}=75(2)^{n-1},$ where $n$ is the number of the year just beginning. Find how many cases there will be at the beginning of the sixth year. Find how many cases there were at the beginning of the first year.

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Problem 44

Solve. See Example 4
A new college had an initial enrollment of 2700 students in $2000,$ and each year the enrollment increases by 150 students. Find the enrollment for each of 5 years, beginning with 2000 .

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Problem 45

An endangered species of sparrow had an estimated population of 800 in 2000 , and scientists predicted that its population would decrease by half each year. Estimate the population in $2004 .$ Estimate the year the sparrow was extinct.

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Problem 46

Solve. See Example 4
A Fibonacci sequence is a special type of sequence in which the first two terms are $1,$ and each term thereafter is the sum of the two previous terms: $1,1,2,3,5,8,$ etc. The formula for the $n$ th Fibonacci term is $a_{n}=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right]$
Verify that the first two terms of the Fibonacci sequence are each 1.

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Problem 47

Sketch the graph of each quadratic function. See Section 8.5
$$
f(x)=(x-1)^{2}+3
$$

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Problem 48

Sketch the graph of each quadratic function. See Section 8.5
$$
f(x)=(x-2)^{2}+1
$$

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Problem 49

Sketch the graph of each quadratic function. See Section 8.5
$$
f(x)=2(x+4)^{2}+2
$$

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Problem 50

Sketch the graph of each quadratic function. See Section 8.5
$$
f(x)=3(x-3)^{2}+4
$$

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Problem 51

Find the distance between each pair of points. See Section 7.3
$$
(-4,-1) \text { and }(-7,-3)
$$

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Problem 52

Find the distance between each pair of points. See Section 7.3
$$
(-2,-1) \text { and }(-1,5)
$$

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Problem 53

Find the distance between each pair of points. See Section 7.3
$$
(2,-7) \text { and }(-3,-3)
$$

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Problem 54

Find the distance between each pair of points. See Section 7.3
$$
(10,-14) \text { and }(5,-11)
$$

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Problem 55

Find the first five terms of each sequence. Round each term after the first to four decimal places.
$$
a_{n}=\frac{1}{\sqrt{n}}
$$

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Problem 56

Find the first five terms of each sequence. Round each term after the first to four decimal places.
$$
\frac{\sqrt{n}}{\sqrt{n}+1}
$$

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Problem 57

Find the first five terms of each sequence. Round each term after the first to four decimal places.
$$
a_{n}=\left(1+\frac{1}{n}\right)^{n}
$$

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Problem 58

Find the first five terms of each sequence. Round each term after the first to four decimal places.
$$
a_{n}=\left(1+\frac{0.05}{n}\right)^{n}
$$

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