Deck 13: Sequences and Series

Question 1

Find the values of x and y for which the sequence
is arithmetic.

Answer 1

Question 2

Dr. Stevens is considering a 30-year mortgage at 4% interest. She can make payments of $3500 a month. What size loan can she afford?

Answer 2

$\$ 733,114.34$

Question 3

Find the amount of an annuity that consists of $30$ annual payments of $\$ 500$ each into an account that pays $7 \%$ interest per year.

Answer 3

$A _ { J } = 500 \frac { ( 1.07 ) ^ { 20 } - 1 } { 0.07 } \approx \$ 47230.39$

Question 4

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $\alpha = 7$
and $r= \frac { 1 } { 7 }$

Question 5

A man gets a job with a salary of $40,000 a year. He is promised an $1800 raise each subsequent year. Find his total earnings for a 10-year period.

Question 6

Write the sum without using sigma notation. Do not evaluate.

Question 7

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $5 + 25 + 125 + \ldots + 15625$

Question 8

A city was incorporated in 2004 with a population of 25,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by the sequence
a) Find the first 5 terms of the sequence
b) Find the population in 2014.

Question 9

Given that the $5 ^ { \text {th } }$ term of an arithmetic sequence is $30$ and the $7 ^ { \text {th } }$ term is $44$ , find the first term.

Question 10

Find the first four terms and the $1000 ^ { \text {th } }$ term of the sequence $c _ { n } = ( - 1 ) ^ { n } n ^ { 2 }$

Question 11

An arithmetic sequence has first term $a = - 2$ and common difference $d = 4$ .
How many terms of this sequence must be added to get $1566$ ?

Question 12

The first four terms of a sequence are given. Determine whether they can be terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic find the common difference. If the sequence is geometric find the common ratio.

Question 13

Find the first four terms and the $10 ^ { \text {th } }$ term of the sequence $a _ { n } = n - 1$

Question 14

Determine whether the expression is a partial sum of a arithmetic or geometric sequence. Then find the sum.

Question 15

Find the
term of the sequence whose first several terms are
.

Question 16

Find the first five terms of the sequence
.

Question 17

How much money should be invested monthly at $6 \%$ per year, compounded monthly, in order to have $\$ 10,000$ in two years?

Question 18

The common ratio of a geometric sequence is $\frac { 3 } { 7 }$ and the fourth term is $\frac { 1 } { 7 }$
Find the third term.

Question 19

Write the sum using sigma notation.

Question 20

Which term of the arithmetic sequence $\frac { 1 } { 2 } , 2 , \frac { 7 } { 2 } , \ldots$ is $23$ ?

Question 21

Find the first four terms and the $1000 ^ { \text {th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

Question 22

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $2 + 4 + 6 + 8 +\dots + 100$

Question 23

A city was incorporated in 2004 with a population of 45,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by the sequence
a) Find the first 5 terms of the sequence
b) Find the population in 2014.

Question 24

Express the repeating decimal $0.345345345 \ldots$
as a fraction.

Question 25

Which term of the arithmetic sequence $\frac { 1 } { 2 } , 2 , \frac { 7 } { 2 } , \ldots$ is $14$ ?

Question 26

Find the coefficient of $- a ^ { \dot { 3 } } b ^ { 3 }$ in the expansion of $( b - a ) ^ { 3 }$

Question 27

Find the $n ^ { \text {th} }$ term of the sequence whose first several terms are $3$ , $9$ , $27$ , $81$ ,.....

Question 28

Write the sum using sigma notation.

Question 29

Find the first five terms of the sequence
, where
.

Question 30

Find the coefficient of $- a ^ { 5 } b ^ { 3 }$ in the expansion of $( b - a ) ^ { 8 }$

Question 31

Find the fourth term and the $n ^ { t h }$ term of the geometric sequence given $a _ { 1 } = 7$ and $\boldsymbol { r } = \frac { 1 } { 7 }$

Question 32

Given that the $50 ^ { \text {th } }$ term of an arithmetic sequence is $259$ and the common difference is $5$ , find the first three terms.

Question 33

Expand the expression
.

Question 34

The first term of a geometric sequence is $12$ and the second term is $4$ .
Find the fifth term.

Question 35

The first four terms of a sequence are given. Determine whether they can be terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic find the common difference. If the sequence is geometric find the common ratio.

Question 36

Write the sum without using sigma notation. Do not evaluate.

Question 37

Find the sum of the infinite geometric series.

Question 38

Find the second term in the expansion of

Question 39

Find the first four terms and the $10 ^ { \mathrm { ct } }$ term of the sequence $a _ { n } = n - 2$

Question 40

A man gets a job with a salary of $50,000 a year. He is promised an $1800 raise each subsequent year. Find his total earnings for a 10-year period.

Question 41

Dr. Stevens is considering a 30-year mortgage at 6% interest. She can make payments of $2500 a month. What size loan can she afford?

Question 42

Find the amount of an annuity that consists of
annual payments of
each into an account that pays
interest per year.

Question 43

How much money must be invested now at
per year, compounded semi-annually, to fund an annuity of
semi-annual payments of
each, the first payment being six months after the initial investment?

Question 44

The first four terms of a sequence are given. Determine whether they can be terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic find the common difference. If the sequence is geometric find the common ratio. $1 , - \frac { 3 } { 2 } , 2 , - \frac { 5 } { 2 }$

A) arithmetic, $d = - \frac { 3 } { 2 }$
B) arithmetic, $d = - \frac { 1 } { 2 }$
C) geomettic, $r = - \frac { 3 } { 2 }$
D) geometric, $r = - \frac { 5 } { 4 }$
E) neither

Question 45

Given that the $50 ^ { \text {th } }$ term of an arithmetic sequence is $259$ and the common difference is $5$ , find the first three terms

A) $15,20,25$
B) $13,18,23$
C) $13,18,24$
D) $14,19,24$
E) $10,15,20$

Question 46

Determine the common ratio, the fifth, and the $n ^ { t h }$ terms of the geometric sequence. $9,9 \sqrt { 3 } , 27,27 \sqrt { 3 } , \dots$

A) $a _ { n } = 9 ( 3 ) ^ { - \cdots }$
B) $a _ { n } = 6 ( \sqrt { 3 } ) ^ { n - 1 }$
C) $a _ { n } = 9 ( \sqrt { 3 } ) ^ { n - 1 }$
D) $a _ { n } = ( \sqrt { 3 } ) ^ { n + 1 }$
E)none

Question 47

Find the fourth term of the geometric sequence given and

A)
B)
C)
D)
E)

Question 48

Find the first four terms of the sequence $a _ { n } = n - 2$

A) $a _ { 1 } = - 1$ , $a _ { 2 } = 0$ , $a _ { 3 } = 1$ , $a _ { 4 } = 2$
B) $a _ { 1 } = 0$ , $a _ { 2 } = 1$ , $a = 2$ , $a _ { 4 } = 3$
C) $a _ { 1 } = - 2$ , $a _ { 2 } = - 3$ , $a _ { 3 } = - 4$ , $a _ { 4 } = - 5$
D) $a _ { 1 } = 1$ , $a _ { 2 } = 2$ , $a = 3$ , $a _ { 4 } = 4$
E) $a _ { 1 } = 2$ , $a _ { 2 } = 3$ , $a _ {3 } = 4$ , $a _ { 4 } = 5$

Question 49

A city was incorporated in 2004 with a population of 25,000. It is expected that the population will increase at a rate of 2% per year. The population n years after 2004 is given by the sequence Find the population in 2014

A)
B)
C)
D)
E) none of these

Question 50

Find the sum of the infinite geometric series.

Question 51

Express the repeating decimal $0.1 \overline { 38 }$ as a fraction.

Question 52

Find the coefficient of $a ^ { 4 } b ^ { 4 }$ in the expansion of $( b - a ) ^ { 3 }$

Question 53

Expand the expression $( 1 - x ) ^ { 3 }$

Question 54

The first term of the arithmetic sequence is $\frac { 2 } { 3 }$ and the common difference is $\left( - \frac { 2 } { 3 } \right\}$ Which term of this sequence is $- \frac { 20 } { 3 }$ ?

A) $10 ^ { \mathrm { th } } \text { term }$
B) $12 ^ { \text {th } } \text { term }$
C) $13 ^ { \text {th } } \text { term }$
D) $16 ^ { \mathrm { th } } \text { term }$
E) $6 ^ { \mathrm { th } } \text { term }$

Question 55

Find the $n ^ { \text {ct } }$ term of the sequence whose first several terms are $\frac { 1 } { 4 } , - \frac { 1 } { 9 } , \frac { 1 } { 16 } , - \frac { 1 } { 25 } , \ldots$

A) $Q _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } }$

B) $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { ( n + 1 ) ^ { 2 } }$

C) $a _ { \mathrm { n } } = \frac { ( n + 1 ) ^ { n } } { 4 ^ { 2 } }$

D) $a _ { n } = \frac { ( 1 ) ^ { n } } { ( 2 n ) ^ { 2 } }$

E) none of these

Question 56

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $5 + 25 + 125 + \dots+ 15625$

Question 57

Find the $1000 ^ { \text {th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

A) $a _ { 1000 } = - \frac { 501 } { 500 }$

B) $a _ { 1000 } = \frac { 251 } { 250 }$

C) $a _ { 1000 } = \frac { 126 } { 125 }$

D) $a _ { 1000 } = - \frac { 126 } { 125 }$

E) $a _ { 1000 } = \frac { 501 } { 500 }$

Question 58

The seventh term of an arithmetic sequence is $- 16$ and the seventeenth term is $- 66$ .
Find the twenty-fourth term.

Question 59

Find the second term in the expansion of

Question 60

Find the first five terms of the sequence $a _ { n } = 3 \left( a _ { n - 1 } + 1 \right)$ , where $a _ { 1 } = 1$

A) $a _ { 1 } = 1$ , $a _ { 2 } = 6$ , $a _ { 3 } = 21$ , $a _ { 4 } = 66$ , $a _ { 3 } = 201$
B) $a _ { 1 } = 3$ , $a _ { 2 } = 9$ , $a _ { 3 } = 27$ , $a _ { 4 } = 69$ , $a _ { 3 } = 226$
C) $a _ { 1 } = 3$ , $a _ { 2 } = 12$ , $a _ { 3 } = 15$ , $a _ { 4 } = 18$ , $a _ { 3 } = 21$
D) $a _ { 1 } = 1$ , $a _ { 2 } = 7$ , $a _ { 3 } = 22$ , $a _ { 4 } = 67$ , $a _ { 3 } = 202$
E) none of these

Question 61

Find the amount of an annuity that consists of annual payments of
Each into an account that pays
Interest per year

A)
B)
C)
D)
E)

Question 62

Write the sum without using sigma notation.

A)
B)
C)
D)
E) none of these

Question 63

Find the first four terms sequence $a _ { n } = n - 1$

A) $a _ { 1 } = - 1$ , $a _ { 2 } = 0$ , $a _ { 3 } = 1$ , $a _ { 4 } = 2$
B) $a _ { 1 } = 0$ , $a _ { 2 } = 1$ , $a = 2$ , $a _ { 4 } = 3$
C) $a _ { 1 } = - 2$ , $a _ { 2 } = - 3$ , $a _ { 3 } = - 4$ , $a _ { 4 } = - 5$
D) $a _ { 1 } = 1$ , $a _ { 2 } = 2$ , $a = 3$ , $a _ { 4 } = 4$
E)none of these

Question 64

Find the second term in the expansion of

A)
B)
C)
D)
E)

Question 65

Dr. Stevens is considering a 30-year mortgage at 4% interest. She can make payments of $3500 a month. What size loan can she afford?

A)
B)
C)
D)
E)

Question 66

Determine whether the expression is a partial sum of an arithmetic or geometric sequence. Then find the sum. $5 + 25 + 125 + \dots + 3125$

A) $19530$
B) $7810$
C) $3125$
D) $3280$
E) $3905$

Question 67

How much money must be invested now at per year, compounded semi-annually, to fund an annuity of
Semi-annual payments of
Each, the first payment being six months after the initial investment?

A)
B)
C)
D)
E)

Question 68

Find the first five terms of the sequence $a _ { n } = 3 a _ { n - 1 } - 1 , \text { where } a _ { 1 } = 3$

A) $a _ { 1 } = 1$ , $a _ { 2 } = 6$ , $a _ { 1 } = 21$ , $a _ { 4 } = 66$ , $a _ { 3 } = 201$
B) $a _ { 1 } = 3$ , $a _ { 2 } = 9$ , $a _ { \jmath } = 27$ , $a _ { 4 } = 69$ , $a _ { 3 } = 226$
C) $a _ { 1 } = 3$ , $a _ { 2 } = 8$ , $a _ { 3 } = 23$ , $a _ { 4 } = 68$ , $a _ { 3 } = 203$
D) $a _ { 1 } = 1$ , $a _ { 2 } = 7$ , $a _ { 3 } = 22$ , $a _ { 4 } = 67$ , $a _ { 3 } = 202$
E) none of these

Question 69

Find the $1000 ^ { \mathrm { th } }$ term of the sequence $a _ { n } = ( - 1 ) ^ { n } \frac { n + 2 } { n }$

A) $a _ { 1000 } = - \frac { 501 } { 500 }$

B) $a _ { 1000 } = \frac { 251 } { 250 }$

C) $a _ { 1000 } = \frac { 126 } { 125 }$

D) $a_{1000}= - \frac { 126 } { 125 }$

E) $a _ { 1000 } = \frac { 501 } { 500 }$

Question 70

Find the sum of the infinite geometric series.

A)
B)
C)
D)
E)

Question 71

Find the term of the sequence whose first several terms are

A)
B)
C)
D)
E) none of these

Question 72

A man gets a job with a salary of $50,000 a year. He is promised an $1800 raise each subsequent year. Find his total earnings for a 10-year period

A)
B)
C)
D)
E)none of these

Question 73

Find the sum. $2 + 4 + 6 + 8 + \dots + 100$

A) $2000$
B) $200$
C) $2050$
D) $1550$
E) none of these

Question 74

Write the sum using sigma notation. $\frac { 1 } { 1 \cdot 2 } + \frac { 1 } { 2 \cdot 3 } + \frac { 1 } { 3 \cdot 4 } + \ldots + \frac { 1 } { 999 \cdot 1000 }$

A) $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n + 1 ) }$

B) $\sum _ { n = 1 } ^ { 1000 } \frac { 1 } { n ( n - 1 ) }$

C) $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n + 1 ) }$

D) $\sum _ { n = 1 } ^ { 1001 } \frac { 1 } { n ( n + 1 ) }$

E) $\sum _ { n = 1 } ^ { 999 } \frac { 1 } { n ( n - 1 ) }$

Question 75

A ball rebounds to one-quarter the height from which it was dropped. Approximate the total vertical distance the ball travels after being dropped from ft above the ground, until it comes to rest

A)
B)
C)
D)
E)

Question 76

Find the coefficient of in the expansion of

A)
B)
C)
D)
E)

Question 77

The first four terms of a sequence are given. Determine whether they can be terms of an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic find the common difference. If the sequence is geometric find the common ratio. $- 5 , - 2 s , - 3 s , - 4 s , K$

A) $\text { arithmetic, } d = - s$
B) $\text { arithmetic, } d = - \frac { 1 } { 2 }$
C) $\text { geometric, } \gamma = \frac { 35 } { 2 }$
D) $\text { geometric, } \gamma = - \frac { 5 } { 4 }$
E) neither

Question 78

The seventh term of an arithmetic sequence is and the tenth term is
) Find the twenty-fourth term

A)
B)
C)
D)
E)

Question 79

Expand the expression.

Question 80

Find the values of a and b for which the sequence is arithmetic

A)
B)
C)
D)
E)