Deck 13: Sequences and Series

Question 1

Is the following sequence arithmetic, geometric, or neither?
$1,3,5,7, \ldots$

Answer 1

arithmetic

Question 2

Is the following sequence arithmetic, geometric, or neither?
$1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \ldots$

Answer 2

neither

Question 3

Is the following sequence arithmetic, geometric, or neither?
$5, \frac{5}{2},-5,-\frac{5}{2}, \ldots$

Answer 3

neither

Question 4

For the arithmetic sequence $a_{1}, a_{2}, \ldots, a_{n}$ , suppose $a_{1}=1, a_{2}=3$ . If $a_{n}=a_{1}+(n-1) d$ , what is $d$ ?

Question 5

For the arithmetic sequence $a_{1}, a_{2}, \ldots, a_{n}$ , suppose $a_{1}=1, a_{2}=5$ . What is $a_{19}$ ?

Question 6

For the arithmetic sequence $a_{1}, a_{2}, \ldots, a_{n}$ , suppose $a_{1}=1, a_{2}=2$ , and the partial sum $S_{1}=1$ . What is $S_{6}$ ?

Question 7

A sequence $a_{n}$ can be defined by a recurrence relation, which gives the first term, $a_{1}$ , and a formula for finding $a_{n}$ in terms of the previous term, $a_{n-1}$ . For the recurrence relation below, what is $a_{4}$ ?
$a_{1}=5 ; a_{n}=\frac{a_{n-1}}{2}$

Question 8

A sequence $a_{n}$ can be defined by a recurrence relation, which gives the first term, $a_{1}$ , and a formula for finding $a_{n}$ in terms of the previous term, $a_{n-1}$ . For the recurrence relation below, what is the formula for the general term?
$a_{1}=1 ; a_{n}=\frac{a_{n-1}}{9}$

A) $a_{n}=\left(\frac{1}{9}\right)^{n}$
B) $a_{n}=\left(\frac{1}{9}\right)^{n-1}$
C) $a_{n}=\frac{1}{9 n}$
D) $a_{n}=\frac{1}{9(n-1)}$

Question 9

Is $1^{3}, 6^{3}, 11^{3}, 16^{3}, \ldots$ an arithmetic sequence?

Question 10

Consider the sequence $1,1+3,1+3+5,1+3+5+7$ , ...involving sums of odd integers.
What is the $36^{\text {th }}$ term of this sequence (i.e. the sum of the first 36 odd numbers)?

Question 11

A person decides to walk for 11 minutes one day, and then increases his walks by 2.5 minutes each day for a month. Let $a_{1}, a_{2}, \cdots, a_{n}$ be the sequence showing the length of time he walks each day, where $a_{n}$ is the length of time he walks on day $n$ . What is $a_{26}$ ?

Question 12

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be an arithmetic sequence with $a_{1}=2$ and $a_{3}=5.8$ . What is $a_{11}$ ?

Question 13

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be an arithmetic sequence with $a_{3}=5$ and $a_{6}=-7$ . What is $a_{10}$ ?

Question 14

Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a geometric sequence with $a_{2}=12$ and $a_{4}=108$ . What is $a_{10}$ ?

Question 15

If $S_{n}$ denotes your total earnings after $n$ weeks of work, is the amount you earned during the $9^{\text {th }}$ week given by $S_{9}-S_{1}$ ?

Question 16

Is the sequence $7,-7,7,-7,7,-7, \ldots$ arithmetic, geometric, neither, or both?

Question 17

Is the sequence $a, 4 a+4,7 a+8,10 a+12, \ldots$ arithmetic, geometric, neither, or both?

Question 18

What is the $12^{\text {th }}$ term of the sequence $1,6,11,16, \ldots$ ? The sequence is either arithmetic or geometric.

Question 19

What is the $9^{\text {th }}$ term of the sequence $1,2,4,8,16, \ldots$ ? The sequence is either arithmetic or geometric.

Question 20

What is the $14^{\text {th }}$ term of the sequence $1,0.5,0,-0.5,-1, \ldots$ ? The sequence is either arithmetic or geometric.

Question 21

What is the sum of the first 22 positive even integers?

Question 22

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $\mathrm{n}$ is the number of years since 1995 (so $S_{1}=3641, S_{2}=3663$ , etc.).

Which of the following gives the change in enrollment between 1999 and 2000 ?

A) $S_{4}$
B) $S_{5}$
C) $a_{4}$
D) $a_{5}$

Question 23

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $\mathrm{n}$ is the number of years since 1995 (so $S_{1}=4449, S_{2}=4478$ , etc.).
$$\begin{array}{ccccccccccc}\text { year } & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\\text { enrollment } & 4449 & 4478 & 4528 & 4575 & 4590 & 4562 & 4598 & 4693 & 4740 & 4773\end{array}$$
Which of the following gives the total enrollment in 2004?

A) $S_{9}$
B) $S_{10}$
C) $a_{9}$
D) $a_{10}$

Question 24

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $n$ is the number of years since 1995 (so $S_{1}=4194, S_{2}=4220$ , etc.).

Which of the following gives the average enrollment between 1998 and 2001 ?

A) $\frac{S_{4}+S_{5}+S_{6}+S_{7}}{4}$
B) $\frac{S_{3}+S_{4}+S_{5}+S_{6}}{4}$
C) $\frac{a_{4}+a_{5}+a_{6}+a_{7}}{4}$
D) $\frac{a_{3}+a_{4}+a_{5}+a_{6}}{4}$

Question 25

The following table shows the enrollment in a private college for the years 1996 to 2005. Interpret these figures as partial sums, $S_{n}$ , of terms of a sequence, $a_{n}$ , where $\mathrm{n}$ is the number of years since 1995 (so $S_{1}=4648, S_{2}=4672$ , etc.).

Which of the following gives the average change in enrollment between 2001 and 2005 ?

A) $\frac{S_{7}+S_{8}+S_{9}+S_{10}}{4}$
B) $\frac{S_{6}+S_{7}+S_{8}+S_{9}}{4}$
C) $\frac{a_{7}+a_{8}+a_{9}+a_{10}}{4}$
D) $\frac{a_{6}+a_{7}+a_{8}+a_{9}}{4}$

Question 26

A person decides to walk for 18 minutes one day, and then increases his walks by 3 minutes each day for a month. Let $a_{1}, a_{2}, \cdots, a_{n}$ be the sequence showing the length of time he walks each day, where $a_{n}$ is the length of time he walks on day $n$ , and let $S_{1}, S_{2}, \cdots, S_{n}$ be the sequence of partial sums. What is $S_{27}$ ?

Question 27

Does $\sum_{n=5}^{13}(-1)^{n-1} n^{2}=25-36+49-64+81-100+121-144+169 ?$

Question 28

If $6+9+12+15+18+21=\sum_{n=1}^{a}(b+c n)$ , then $a=\ldots, b=\ldots$ , and $c=$

Question 29

What is $\sum_{k=1}^{30}(3 k+2) ?$

Question 30

Does $\sum_{i=1}^{17} 7 i+\sum_{j=3}^{17}-7 j=0$ ?

Question 31

A child building a tower with blocks places 17 blocks in the first row, 16 blocks in the second row, 15 blocks in the third row, and so forth. How many blocks are in the tower if it has 14 rows total?

Question 32

How many terms are there in the series $\sum_{n=5}^{22}(-1)^{n-1} 6 n^{2}$ ?

Question 33

If $S_{n}$ denotes your total earnings after $n$ weeks of work, what is the average amount of money you earned each week for the first 14 weeks?

A) $\sum_{i=1}^{14} S_{i}$
B) $\frac{1}{14} \sum_{i=1}^{14} S_{i}$
C) $14 S_{14}$
D) $S_{14} / 14$

Question 34

The time it takes an employee to assemble a bicycle is 57 minutes the first time he does it. The second time, it take him 56 minutes; the third time it takes 55 minutes, and so on. If this trend continues, how many minutes will it take him to assemble the $11^{\text {th }}$ bicycle?

Question 35

The time it takes an employee to assemble a bicycle is 57 minutes the first time he does it. The second time, it take him 56 minutes; the third time it takes 55 minutes, and so on. If this trend continues and there are 11 bicycles to assemble, it should take him a total of ---------- hours and -----------minutes.

Question 36

Which of the following is the correct expansion of $\sum_{i=2}^{5}(i+4)^{2}$ ?

A) $6^{2}+7^{2}+8^{2}+9^{2}$
B) $2^{2}+3^{2}+4^{2}+5^{2}$
C) $\left(2^{2}+4\right)+\left(3^{2}+4\right)+\left(4^{2}+4\right)+\left(5^{2}+4\right)$
D) $5^{2}+6^{2}+7^{2}+8^{2}$

Question 37

Evaluate the following sum :
$10+13+16+19+\ldots+46$

Question 38

Find the sum of the first 400 odd integers.

Question 39

Write $11+16+21+\ldots+56$ using sigma notation.

Question 40

Write $6-1-8-\ldots-78$ using sigma notation.

Question 41

If you were to fill out a pedigree chart, the first column would have one name (yours), the second column would have two names (your parents), the third column would have 4 names (your grandparents), etc. How many names would be on the chart if there were 9 columns total?

Question 42

The repeating decimal 0.757575 . . can be written as an infinite geometric series $0.75+$ $0.0075+0.000075+\ldots$ Use this fact to find a fraction that is equivalent to 0.757575 . ... Your answer should be of the form "A/B".

Question 43

If $1-2+4-8+16-\cdots+256$ is a geometric series, give its ratio. If not, enter "not geometric"

Question 44

Find $\sum_{n=0}^{7} 6\left(\frac{1}{5}\right)^{n}$ to 3 decimal places.

Question 45

If $2+10+50+250+1,250+6,250=\sum_{n=0}^{a} b(c)^{n}$ , then $a=\ldots, b=\ldots$ , and $c=$

Question 46

Is the series $1^{2}+8^{2}+15^{2}+22^{2}+\cdots$ arithmetic, geometric, or neither?

Question 47

Is the series $3.6+4.7+3.6+4.7+\cdots$ arithmetic, geometric, or neither?

Question 48

Is $x+\frac{x^{2}}{5}+\frac{x^{3}}{10}+\frac{x^{4}}{15}+\cdots$ a geometric series?

Question 49

Find the sum of the series $5+\frac{5}{3}+\frac{5}{9}+\frac{5}{27}+\cdots$ . Round to 3 decimal places if necessary.

Question 50

A new employee is offered two salary packages. Both start with an initial salary of $43,000. Package A offers a $2000 increase each year. Package B offers a 4% increase each year. After 11 years, the annual salary with package A would be ---------$ and the annual salary with package B would be$ -----------.Round both answers to the nearest dollar.

Question 51

Find the sum of the series $\sum_{k=0}^{\infty} \frac{3^{k}+2}{4^{k}}$ . Round to 2 decimal places.

Question 52

Does $6(0.6)^{5}+6(0.6)^{6}+\cdots+6(0.6)^{15}=\sum_{k=5}^{15} 6(0.6)^{k}$ ?

Question 53

Evaluate $3(0.3)^{4}+3(0.3)^{5}+\cdots+3(0.3)^{24}$ to 3 decimal places.

Question 54

To save for their child's college education, parents put $\$ 100$ at the beginning of each month into an account that pays $4 \%$ annual interest, compounded monthly. How much will they have saved if they do this for 18 years? Round to the nearest cent.

Question 55

To save for their child's college education, parents put $\$ 200$ at the beginning of each month into an account that pays $6.5 \%$ annual interest, compounded monthly. They do this for 18 years. How much money per month would they have had to save if they needed the same total amount in 18 years, but their account only paid $4.5 \%$ annual interest, compunded monthly? Round to the nearest cent.

Question 56

A patient takes $600 \mathrm{mg}$ of a pain-killing drug twice a day. Every 12 hours, the patient's body metabolizes $30 \%$ of the drug present. How many mg of the drug remain in the patient's body after 8 days (right after the $16^{\text {th }}$ dose)? Round to 2 decimal places.

Question 57

A patient takes $400 \mathrm{mg}$ of a pain-killing drug twice a day. Every 12 hours, the patient's body metabolizes $25 \%$ of the drug present. If the patient stops taking the drug after 1 week, how many $\mathrm{mg}$ of the drug remain in the patient's body 2 days after that? Round to 2 decimal places.

Question 58

The population of Arizona rose from 5,130,635 in 2000 to 5,939,292 in 2005.
Assuming a constant percent annual growth rate, what is the population expected to be in 2021? Round to the nearest whole number.

Question 59

The population of Arizona rose from 5,130,635 in 2000 to 5,939,292 in 2005. Assuming a constant percent growth rate, in what year is the population first expected to exceed 12 million?

Question 60

Does $\sum_{i=1}^{15} i^{2}-\sum_{j=1}^{16} j^{2}=-256$ ?

Question 61

Does $\sum_{i=1}^{27}(i+5)^{2}=\sum_{j=6}^{32} j^{2}$ ?

Question 62

What is the present value of a lottery prize of 1 million dollars, paid out at a rate of $\$ 20,000$ per year for 50 years? Assume an annual interest rate of $6 \%$ . Round to the nearest dollar.

Question 63

A patient takes $400 \mathrm{mg}$ of a pain-killing drug twice a day. Every 12 hours, the patient's body metabolizes $30 \%$ of the drug present. If the patient continues to take the drug indefinitely, how many $\mathrm{mg}$ of the drug will remain in the patient's body in the long run? Round to 2 decimal places.

Question 64

Find the sum of the series $1-\frac{x}{6}+\frac{x^{2}}{36}-\frac{x^{3}}{216}+\cdots$ , for $|x| \leq 6$ .

A) $\frac{6}{6+x}$
B) $\frac{6}{6-x}$
C) $\frac{1}{6+x}$
D) $\frac{1}{6-x}$

Question 65

What is $1-\frac{1}{8}+\frac{1}{64}-\frac{1}{512}+\cdots$ ? Round to 2 decimal places.

Question 66

Subtract the following two equations and solve for $S$ :
$\begin{gathered}S=a+a x+a x^{2}+a x^{3}+\cdots+a x^{12} \\X S=a x+a x^{2}+a x^{3}+a x^{4}+\cdots+a x^{13}\end{gathered}$
Assume $x$ is not equal to 1 .

A) $S=\frac{a\left(1-x^{13}\right)}{x}$
B) $S=\frac{a\left(1-x^{13}\right)}{1-x}$
C) $S=-a\left(1-x^{13}\right)$
D) $S=a\left(1-x^{13}\right)$

Question 67

Let $S=5+5 x+5 x^{2}+5 x^{3}+\cdots+5 x^{n}$ , with $|x|<1$ . As $n \rightarrow \infty$ , what happens to $S$ ?

A) $S \rightarrow 5$
B) $S \rightarrow \frac{5}{x}$
C) $S \rightarrow \frac{5}{1-x}$
D) $S \rightarrow \frac{5}{1+x}$

Question 68

An athlete is offered a signing bonus of $\$ 2.5$ million per year for 10 years. If the interest rate is $5.8 \%$ per year, compounded annually, what is the present value of this bonus? Round to the nearest dollar.

Question 69

A plaintiff is awarded a settlement of $\$ 15,000$ per year (paid at the beginning of the year) for the rest of his life. Assuming 3.6\% annual interest, how much money would the defendant have to put aside now in order to ensure that the settlement would be paid (no matter how many years the plaintiff lived)? In other words, what is the present value of the settlement if the plaintiff may live indefinitely? Round to the nearest dollar.

Question 70

Does the infinite geometric series $20-16+12.8-10.24+\cdots$ converge or diverge?

Question 71

Write $0.151515 \ldots$ as a fraction.

Question 72

An employee is offered a salary of $\$ 100,000$ per year for 6 years. What is the present value of the contract on the date of the first payment if the interest rate is $2.3 \%$ per year, compounded continuously. Round your answer to two decimal places.

Question 73

Write $3.368368 \ldots$ as a fraction or improper fraction.

Question 74

What is the present value of a $\$ 1000$ bond which pays $\$ 50$ each year for 15 years, with the first payment one year from now? Assume the interest rate is 5.2\% per year compounded annually.

Question 75

What is the present value of a $\$ 1000$ bond which pays $\$ 40$ each year for 15 years, with the first payment one year from now? Assume the interest rate is 5.5\% per year compounded annually.

A) $\$ 849.44$
B) $\$ 401.50$
C) $\$ 856.16$
D) $\$ 843.07$

Question 76

Find the sum $\sum_{i=3}^{\infty} \frac{4^{i}+2}{5^{i}}$ . Round your answer to 3 decimal places.

Question 77

Your doctor tells you to take $38 \mathrm{mg}$ of a certain drug every 12 hours. It is known that only $6 \%$ of the drug is still in the body at the end of 12 hours. What quantity of the drug is in the body right after taking the 11th dose? Give your answer correct to 3 decimal places.

Question 78

Your doctor tells you to take $36 \mathrm{mg}$ of a certain drug every 12 hours. It is known that only $6 \%$ of the drug is still in the body at the end of 12 hours. Assuming you continue to take the drug, what happens to the drug level in the long run? Give your answer correct to 3 decimal places.

Question 79

For what values of $x$ does the sum $\sum_{i=1}^{\infty}(3 x-5)^{3 i}$ converge?

Question 80

For what values of $x$ does the sum $\sum_{i=5}^{\infty}\left(\frac{1}{4} x\right)^{i}$ converge?