Deck 16: Series and Taylor Polynomials Web

Question 1

Find the indicated term of the sequence. $$\begin{array} { l } a _ { n } = ( - 1 ) ^ { n } ( 3 n - 1 ) \\a _ { 16 } = \square\end{array}$$

A)-44
B)47
C)49
D)-2
E)45

Answer 1

47

Question 2

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) $\alpha _ { n } = - 2 - 7 n$

A)7
B)2
C)-2
D)-7
E)not arithmetic

Answer 2

-7

Question 3

Determine whether the sequence is arithmetic. If so, find the common difference. 2, 1, 0, -1, -2

A)3
B)-1
C)2
D)1
E)not arithmetic

Answer 3

-1

Question 4

Determine whether the sequence is arithmetic. If so, find the common difference. 3, 9, 27, 81, 243

A)3
B)3n
C)3n - 3n-1
D)-3
E)not arithmetic

Question 5

Write an expression for the most apparent nth term of the sequence. (Assume that $n$ begins with 1.) $\frac { 1 } { 3 } , - \frac { 1 } { 9 } , \frac { 1 } { 27 } , - \frac { 1 } { 81 } , \frac { 1 } { 243 } , \mathrm {~K}$

A) $a _ { n } = \frac { ( - 1 ) ^ { n } } { 3 ^ { n } }$
B) $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 3 ^ { n } }$
C) $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 2 ^ { n } }$
D) $a _ { n } = \frac { ( - 1 ) ^ { n } } { 2 ^ { n } }$
E) $a _ { n } = \frac { 1 } { 2 ^ { n } }$

Question 6

A deposit of $2000 is made in an account that earns 6% interest compounded monthly. The balance in the account after n months is given by $A _ { n } = 2000 \left( 1 + \frac { 0.06 } { 12 } \right) ^ { n } , n = 1,2,3 , \mathrm {~K}$ Find the balance in the account after 11 years by finding the 132th term of the sequence. Round to the nearest penny.

A)$4,379,295.09
B)$3863.23
C)$3844.01
D)$265,320.00
E)$3882.54

Question 7

Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 9, 12, 15, 18, 21

A) $a _ { n } = 6 n + 3$
B) $a _ { n } = ( - 1 ) ^ { n } ( 3 n + 6 )$
C) $a _ { n } = 3 n$
D) $a _ { n } = 3 ^ { n } + 6$
E) $a _ { n } = 3 n + 6$

Question 8

Write the first five terms of the sequence. (Assume that n begins with 1.) $$a _ { n } = 3 n + 7$$

A)-4, -1, 2, 5, 8
B)7, 10, 13, 16, 19
C)10, 6, 9, 12, 15
D)10, 13, 16, 19, 22
E)10, 17, 24, 31, 38

Question 9

Find the fifth term of the sequence that has the given nth term. $$a _ { n } = \frac { ( n + 2 ) ! } { n ! }$$

A) $42$
B) $30$
C) $12$
D) $20$
E) $6$

Question 10

Evaluate the series. $\sum _ { i = 1 } ^ { 4 } ( 4 i + 3 ) ( 3 i - 4 )$

A) $15$
B) $242$
C) $873$
D) $495$
E) $2100$

Question 11

Suppose the ratio $a _ { n }$ of alligators to pythons in a marshland from 2001 to 2008 can be approximated by the model $a _ { n } = 24 - 3.5 n + 0.34 n ^ { 2 }$ $n = 1,2 , \mathrm {~K} , 8$ where $n$ is the year, with $n = 1,2 , \mathrm {~K} , 8$ corresponding to $2001,2002 , \mathrm {~K} , 2008$ In 2006, the total number of alligators and pythons in the marsh was about 900. In that year, how many were pythons?

A)848
B)55
C)844
D)52
E)845

Question 12

Match the sequence with the graph of its first 10 terms. $a _ { n } = \frac { 3 n } { n + 1 }$

A)
B)
C)
D)
E)

Question 13

Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) $$- 5 - \frac { 2 } { 1 } , - 5 - \frac { 2 } { 2 } , - 5 - \frac { 2 } { 3 } , - 5 - \frac { 2 } { 4 } , - 5 - \frac { 2 } { 5 } ,$$

A) $a _ { n } = - 5 - \frac { n } { 2 }$
B) $a _ { n } = 2 - \frac { - 5 } { n }$
C) $a _ { n } = 2 - \frac { n } { - 5 }$
D) $a _ { n } = - 5 - \frac { 2 } { n }$
E) $a _ { n } = - 5 - \frac { 2 } { n + 1 }$

Question 14

Find a formula for an for the arithmetic sequence. $a _ { 3 } = 8 , a _ { 7 } = 36$

A) $a _ { n } = - 6 + 7 n$
B) $a _ { n } = 13 - 6 n$
C) $a _ { n } = 7 - 6 n$
D) $a _ { n } = - 6 ( 7 ) ^ { n }$
E) $a _ { n } = - 13 + 7 n$

Question 15

Suppose that the annual payroll $a _ { n }$ (in billions of dollars) of new car dealerships in the United States from 2000 to 2005 can be approximated by the model $a _ { n } = 44.7 + 1.51 n - 0.108 n ^ { 2 },$ $n = 0,1,2,3,4,5$ where $n$ represents the year, with $n = 0$ corresponding to 2000. Find the total payroll from 2000 to 2005 by evaluating the sum $\sum _ { n = 0 } ^ { 5 } \left( 44.7 + 1.51 n - 0.108 n ^ { 2 } \right)$ Round your answer to the nearest ten million dollars.

A)$334.78 billion
B)$284.91 billion
C)$377.86 billion
D)$290.08 billion
E)$235.36 billion

Question 16

Write the given series in summation notation. $\frac { 1 } { 5 } + \frac { 1 } { 40 } + \frac { 1 } { 135 } + \frac { 1 } { 320 } + \frac { 1 } { 625 } + \frac { 1 } { 1080 }$

A) $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 4 } }$
B) $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 3 } }$
C) $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 5 } }$
D) $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 3 } }$
E) $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 4 } }$

Question 17

Simplify the factorial expression. $\frac { 14 ! } { 12 ! }$

A)2184
B)182
C) $\frac { 7 } { 6 }$
D)14
E)2730

Question 18

Find the sum. $\sum _ { i = 1 } ^ { 3 } ( 3 i - 5 )$

A)-1
B)-2
C)3
D)4
E)18

Question 19

Write the given series in summation notation. $2 - 4 + 8 - 16 + 32 - 64 + 128$

A) $\sum _ { i = 1 } ^ { 7 } 4 ^ { i } ( - 1 ) ^ { i }$
B) $\sum _ { i = 1 } ^ { 7 } 4 ^ { i } ( - 1 ) ^ { i + 1 }$
C) $\sum _ { i = 1 } ^ { 7 } 2 ^ { i } ( - 1 ) ^ { i + 1 }$
D) $\sum _ { i = 1 } ^ { 7 } 2 ^ { i } ( - 1 ) ^ { i }$
E) $\sum _ { i = 1 } ^ { 7 } 3 ^ { i } ( - 1 ) ^ { i }$

Question 20

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that n begins with 1.) $a _ { n } = - 8 \left( \frac { 1 } { 5 } \right) ^ { n }$

A)-8
B)5
C) $\frac { 1 } { 5 }$
D)8
E)not arithmetic

Question 21

The annual sales $a _ { n }$ (in millions of dollars) for a certain company from 2001 to 2006 can be approximated by the model $a _ { n } = 500.7 + 161.52 n,$ $n = 0,1 , K , 5$ where $n$ represents the year, with $n = 0$ corresponding to 2001. Find the total sales from 2001 to 2004. Round to the nearest million.

A)$2972 million
B)$4119 million
C)$1987 million
D)$2229 million
E)$1502 million

Question 22

Find the sum of the integers from 5 to 27.

A)378
B)22
C)736
D)368
E)756

Question 23

Determine whether the sequence is geometric. If so, find the common ratio. 1, -3, 9, -27, ...

A)-3
B)1
C) $- \frac { 1 } { 3 }$
D)3
E)not geometric

Question 24

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

A) $79$
B) $89$
C) $84$
D) $59$
E) $54$

Question 25

Use summation notation to write the sum below. $2 + 4 + 6 + 8 + 10 + \mathrm { K } + 200$

A) $\sum _ { n = 1 } ^ { 150 } 5 n$
B) $\sum _ { n = 1 } ^ { 100 } 4 n$
C) $\sum _ { n = 1 } ^ { 150 } 4 n$
D) $\sum _ { n = 1 } ^ { 150 } 2 n$
E) $\sum _ { n = 1 } ^ { 100 } 2 n$

Question 26

Match the arithmetic sequence with its graph from the choices below. $a _ { n } = 35 - 3 n$

A)
B)
C)
D)
E)

Question 27

Find a formula for $a _ { n }$ for the arithmetic sequence below. $- 2 , - 1,0,1,2 , \mathbf { K }$

A) $a _ { n } = 3 n + 1$
B) $a _ { n } = - 3 n - 1$
C) $a _ { n } = - n - 3$
D) $a _ { n } = n - 3$
E) $a _ { n } = n + 3$

Question 28

Find the indicated nth term of the geometric sequence. 7th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }$

A) $\frac { 4 } { 2187 }$
B) $\frac { 3 } { 4096 }$
C) $\frac { 4 } { 6561 }$
D) $\frac { 4 } { 729 }$
E) $\frac { 4 } { 243 }$

Question 29

Determine whether the sequence is geometric. If so, find the common ratio. 5, 7, 9, 11, ...

A)2
B)5
C) $\frac { 1 } { 2 }$
D)-2
E)not geometric

Question 30

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

A)181
B)188
C)160
D)159.4
E)160.6

Question 31

A heavy object (with negligible air resistance) is dropped from a plane. During the first second of fall, the object falls 17.4 meters; during the second second, it falls 52.2 meters; during the third second, it falls 87.0 meters; and during the fourth second, it falls 121.8 meters. If this pattern continues, how many meters will the object fall in 10 seconds?

A)2505.6 meters
B)1409.4 meters
C)626.4 meters
D)1740.0 meters
E)852.6 meters

Question 32

Find the partial sum. $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )$

A)-39,339
B)-38,781
C)-39,620
D)-39,903
E)-39,340

Question 33

Match the geometric sequence with its graph from the choices below. $a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }$

A)
B)
C)
D)
E)

Question 34

Use a graphing utility to graph the first 10 terms of the sequence. $a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }$

A)
B)
C)
D)
E)

Question 35

Find the nth term of the geometric sequence. $- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }$

A) $2 \left( - \frac { 5 } { 4 } \right) ^ { n - 1 }$
B) $2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$
C) $- 2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$
D) $- 2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$
E) $2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$

Question 36

Write the first five terms of the arithmetic sequence. $a _ { 5 } = 19 , a _ { 15 } = 79$

A)-5, -11, -17, -23, -29
B)1, 7, 13, 19, 25
C)-5, 1, 7, 13, 19
D)-5, -30, -180, -1080, -6480
E)-5, 1, -4, -9, -14

Question 37

Write the first five terms of the geometric sequence. $a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }$

A)-1, -7, -13, -19, -25
B)-1, 6, -36, 216, -1296
C) $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \frac { 1 } { 7776 }$
D) $1 , - 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 }$
E) $- 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 }$

Question 38

Find the nth term of the geometric sequence. $3 , - 12,48 , \mathrm {~K}$

A) $- 4 ( 3 ) ^ { n - 1 }$
B) $4 ( 3 ) ^ { n - 1 }$
C) $- 3 ( 4 ) ^ { n - 1 }$
D) $4 ( - 3 ) ^ { n - 1 }$
E) $3 ( - 4 ) ^ { n - 1 }$

Question 39

Logs are stacked so that there are 17 logs in the bottom row, 16 logs in the second row from the bottom, and so on, decreasing by 1 log each row. How many logs are there in the first five rows from the bottom?

A)117 logs
B)108 logs
C)75 logs
D)87 logs
E)125 logs

Question 40

Consider a job offer with a starting salary of $43,200 and a given annual raise of $2175. Determine the total compensation from the company through seven full years of employment.

A)$406,500
B)$291,825
C)$185,850
D)$136,125
E)$348,075

Question 41

Write an expression for the nth term of the sequence $\frac { 1 } { 3 } , \frac { 2 } { 9 } , \frac { 4 } { 27 } , \frac { 8 } { 81 } , \mathrm {~K}$ .

A) $a _ { n } = \frac { 2 ^ { n - 1 } } { 3 ^ { n } }$
B) $a _ { n } = \frac { 2 ^ { n + 1 } } { 3 ^ { n } }$
C) $a _ { n } = \frac { 2 ^ { n } } { 3 }$
D) $a _ { n } = \frac { 6 ^ { n + 1 } } { 4 ^ { n - 1 } }$
E) $a _ { n } = \frac { 6 ^ { n - 1 } } { 4 ^ { n } }$

Question 42

Find the sum of the finite geometric series. Round to the nearest hundredth. $\sum _ { n = 1 } ^ { 6 } \left( - \frac { 8 } { 9 } \right) ^ { n }$

A) $- 0.24$
B) $- 0.10$
C) $- 0.33$
D) $- 0.80$
E) $- 0.63$

Question 43

Find the rational number representation of the repeating decimal. $0 . \overline { 437 }$

A) $\frac { 437 } { 9999 }$
B) $\frac { 43.7 } { 999 }$
C) $\frac { 437 } { 9 }$
D) $\frac { 437 } { 999 }$
E) $\frac { 437 } { 99 }$

Question 44

Write the first five terms of the sequence. an = $\left( - \frac { 5 } { 6 } \right) ^ { n }$

A) $\frac { 5 } { 6 } , - \frac { 25 } { 36 } , \frac { 125 } { 216 } , - \frac { 625 } { 1296 } , \frac { 3125 } { 7776 }$
B) $- \frac { 5 } { 6 } , \frac { 25 } { 36 } , - \frac { 125 } { 216 } , \frac { 625 } { 1296 } , - \frac { 3125 } { 7776 }$
C) $\frac { 5 } { 6 } , \frac { 25 } { 36 } , \frac { 125 } { 216 } , \frac { 625 } { 1296 } , \frac { 3125 } { 7776 }$
D) $- \frac { 5 } { 6 } , - \frac { 25 } { 36 } , - \frac { 125 } { 216 } , - \frac { 625 } { 1296 } , - \frac { 3125 } { 7776 }$
E)none of the above

Question 45

Determine the convergence or divergence of the sequence $9 - \frac { 4 } { 5 ^ { n } }$ . If the sequence converges, use a symbolic algebra utility to find its limit.

A)9
B)4
C)5
D) $- \infty$
E)The sequence diverges.

Question 46

Find the sum of the finite geometric series. $\sum _ { n = 1 } ^ { 4 } 3 ( - 3 ) ^ { n }$

A) $180$
B) $- 63$
C) $1638$
D) $132,858$
E) $- 549$

Question 47

Use summation notation to write the sum. $4 - 8 + 16 - K + 64$

A) $\sum _ { n = 0 } ^ { 4 } 4 ( - 2 ) ^ { n - 1 }$
B) $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n }$
C) $\sum _ { n = 1 } ^ { 5 } 4 ( - 2 ) ^ { n - 1 }$
D) $\sum _ { n = 1 } ^ { 3 } 4 ( - 2 ) ^ { n - 1 }$
E) $\sum _ { n = 1 } ^ { 4 } 4 ( - 2 ) ^ { n + 1 }$

Question 48

Write an expression for the nth term of the sequence 2, 8, 26, 80, ....

A) $a_ { n } = 1 - 3 ^ { n }$
B) $a _ { n } = 4 ^ { n } + 1$
C) $a _ { n } = 3 ^ { n } - 1$
D) $a _ { n } = 4 ^ { n } - 5$
E) $\alpha _ { n } = 1 + 3 ^ { n }$

Question 49

Find the limit of the sequence $a _ { n } = \frac { 1 } { n ^ { 9 / 2 } }$ .

A) $\infty$
B) $1$
C) $0$
D) $\frac { 1 } { 2 }$
E)The sequence diverges.

Question 50

Write the first five terms of the sequence. an = $2 - \frac { 5 } { n } - \frac { 7 } { n ^ { 2 } }$

A) $- 10 , - \frac { 9 } { 4 } , - \frac { 4 } { 9 } , \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$
B) $- 10 , - \frac { 13 } { 2 } , - \frac { 16 } { 3 } , - \frac { 19 } { 4 },$ $- \frac { 22 } { 5 }$
C) $- 10 , - \frac { 19 } { 4 } , - \frac { 34 } { 9 } , - \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$
D) $- 10 , \frac { 19 } { 4 } , - \frac { 34 } { 9 } , \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$
E) $- 10 , \frac { 9 } { 4 } , - \frac { 4 } { 9 } , - \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$

Question 51

Find the limit of the following sequence. $a _ { n } = \frac { n ^ { 2 } - 25 } { n + 5 }$

A) $\infty$
B) $0$
C) $0.5$
D) $- \infty$
E)The sequence diverges.

Question 52

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 7 } 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }$

A) $\frac { 390,369 } { 31,250 }$
B) $\frac { 192 } { 5 }$
C) $25,999$
D) $\frac { 5187 } { 3125 }$
E) $\frac { 77,997 } { 15,625 }$

Question 53

Find the limit of the following sequence. $a _ { n } = \frac { ( n - 2 ) ! } { n ! }$

A)3
B)1
C) $0$
D)-3
E)The sequence diverges.

Question 54

The annual profit $a _ { n }$ (in millions of dollars) for a certain company from 2000 to 2005 can be approximated by the model $a _ { n } = 302.58 e ^ { 0.196 n },$ $n = 0,1 , K , 5$ where $n$ represents the year, with $n = 0$ corresponding to 2000. Use the formula for the sum of a finite geometric sequence to approximate the total profit earned during this six-year period. Round to the nearest ten-thousand dollars.

A)$3810.36 million
B)$4112.94 million
C)$3132.16 million
D)$2829.58 million
E)$2325.95 million

Question 55

Write the rational number $0 . \overline { 81 }$ as the quotient of two integers in simplest form.

A) $\frac { 9 } { 11 }$
B) $\frac { 4 } { 11 }$
C) $\frac { 5 } { 11 }$
D) $\frac { 10 } { 11 }$
E) $\frac { 8 } { 11 }$

Question 56

Find the sum of the infinite geometric series. $\sum _ { n = 1 } ^ { \infty } \left( - \frac { 1 } { 3 } \right) ^ { n }$

A) $\frac { 3 } { 2 }$
B) $\frac { 3 } { 4 }$
C) $- \frac { 1 } { 4 }$
D) $- \frac { 1 } { 2 }$
E) $- \frac { 3 } { 4 }$

Question 57

Write an expression for the nth term of the sequence. $9 , - \frac { 9 } { 4 } , \frac { 9 } { 9 } , - \frac { 9 } { 16 } , \mathrm {~L}$

A) $\frac { 9 - 1 ^ { n + 1 } } { n ^ { 2 } }$
B) $\frac { 9 ( - 1 ) ^ { n } } { n ^ { 2 } }$
C) $\frac { 9 ( - 1 ) ^ { n - 1 } } { n ^ { 2 } }$
D) $\frac { - 1 ^ { n - 1 } } { 9 n ^ { 2 } }$
E) $\frac { ( - 1 ) ^ { n } } { 9 n ^ { 2 } }$

Question 58

Find the sum of the infinite geometric series below. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { 8 } \right) ^ { n - 1 }$

A) $\frac { 11 } { 10 }$
B) $\frac { 8 } { 7 }$
C) $\frac { 3 } { 2 }$
D) $\frac { 10 } { 9 }$
E) $\frac { 6 } { 5 }$

Question 59

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } 4 \left( - \frac { 1 } { 2 } \right) ^ { n }$

A) $- \frac { 8 } { 3 }$
B) $\frac { 8 } { 3 }$
C) $\frac { 4 } { 3 }$
D) $- \frac { 4 } { 3 }$
E)undefined

Question 60

Find the limit of the following sequence. $a _ { n } = 1 + ( - 1 ) ^ { n }$

A) $\infty$
B)1
C)2
D) $- \infty$
E)The sequence diverges.

Question 61

The repeating decimal $0 . \overline { 2 }$ is expressed as a geometric series $0.2 + 0.02 + 0.002 + 0.0002 + \ldots$ . Write the decimal $0 . \overline { 2 }$ as the ratio of two integers.

A) $\frac { 2 } { 99 }$
B) $\frac { 7 } { 33 }$
C) $\frac { 2 } { 9 }$
D) $\frac { 5 } { 11 }$
E) $\frac { 9 } { 2 }$

Question 62

A deposit of $\$$ 200 is made each month in an account that earns 8.4% interest, compounded monthly. The balance in the account after n months is given by $A _ { n } = 200 ( 201 ) \left[ ( 1.007 ) ^ { n } - 1 \right]$ . Find the balance after 22 years by computing the 264th term of the sequence. Round your answer to two decimal places.

A)$213,316.53
B)$293,716.53
C)$6,667.78
D)$86,634.61
E)$281.40

Question 63

A ball is dropped from a height of 14 feet, and on each rebound it rises to $\frac { 2 } { 5 }$ its preceding height. Write an expression for the height of the nth rebound.

A) $h _ { n } = \frac { \left( \frac { 2 } { 5 } \right) ^ { n } } { 14 }$
B) $h _ { n } = 14 \left( \frac { 2 } { 5 } \right) ^ { n }$
C) $h _ { n } = 14 \left( \frac { 5 } { 2 } \right) ^ { n }$
D) $h _ { n } = \frac { 14 } { \left( \frac { 2 } { 5 } \right) ^ { n } }$
E) $h _ { n } = \left( 14 \frac { 5 } { 2 } \right) ^ { n }$

Question 64

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 8 } { 9 } \right) ^ { n }$

A) $81$
B) $63$
C) $72$
D)9
E)8

Question 65

Determine the convergence or divergence of the following series. Use a symbolic algebra utility to verify your result. $\sum _ { n = 0 } ^ { \infty } \frac { 4 } { 2 ^ { n } }$

A)The series diverges.
B)The series converges.

Question 66

Determine whether the series $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a p-series.

A) $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is not a $p -$ series.
B) $\sum _ { n = 1 } ^ { \infty } n ^ { - 4 / 5 }$ is a $p -$ series.

Question 67

What are the next three terms in the arithmetic sequence $7,3 , - 1 , \ldots$ ?

A) $- 5 , - 9 , - 13$
B) $11,15,19$
C) $20 , - 80,320$
D) $28,35,42$
E) $- 1 , - 5 , - 9$

Question 68

Determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } ( 0.650 ) ^ { n }$ . Use a symbolic algebra utility to verify your result.

A)The series converges.
B)The series diverges.

Question 69

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { 900 }$ . Use a symbolic algebra utility to verify your result.

A)The series converges.
B)The series diverges.

Question 70

Find the next three terms of the geometric sequence. $5,15,45 , \ldots$

A) $55,65,75 , \ldots$
B) $135,405,1215 , \ldots$
C) $75,105,135 , \ldots$
D) $48,51,54 , \ldots$
E) $45,135,405 , \ldots$

Question 71

Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] $0 . \overline { 48 }$

A) $\frac { 49 } { 99 }$
B) $\frac { 49 } { 100 }$
C) $\frac { 4 } { 5 }$
D) $\frac { 16 } { 33 }$
E) $\frac { 12 } { 25 }$

Question 72

Write the first five terms of the sequence of partial sums. $2 + \frac { 2 } { 4 } + \frac { 2 } { 9 } + \frac { 2 } { 16 } + \frac { 2 } { 25 } + L$

A) $2 , \frac { 5 } { 3 } , \frac { 49 } { 16 } , \frac { 205 } { 66 } , \frac { 5269 } { 1600 }$
B) $2 , \frac { 1 } { 2 } , \frac { 2 } { 9 } , \frac { 1 } { 8 } , \frac { 2 } { 25 }$
C) $2 , \frac { 5 } { 2 } , \frac { 49 } { 18 } , \frac { 205 } { 72 } , \frac { 5269 } { 1800 }$
D) $2 , \frac { 7 } { 2 } , \frac { 55 } { 18 } , \frac { 215 } { 72 } , \frac { 53 } { 18 }$
E) $2 , \frac { 1 } { 1 } , \frac { 45 } { 16 } , \frac { 100 } { 33 } , \frac { 105 } { 32 }$

Question 73

The annual spending by tourists in a resort city is 200 million dollars. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the resort city. If this pattern continues, write the geometric series that gives the total amount of spending generated by the 200 million dollars (including the initial outlay of 200 million dollars) and find the sum of the series.

A)The geometric series is $\sum _ { n = 1 } ^ { \infty } 200 ( 75 ) ^ { n + 1 }$ .The sum of the series is $\$$ 800.00 million.
B)The geometric series is $\sum _ { n = 1 } ^ { \infty } 200 ( 0.25 ) ^ { n }$ .The sum of the series is $\$$ 15,000 million.
C)The geometric series is $\sum _ { n = 0 } ^ { \infty } 200 ( 0.75 ) ^ { n }$ .The sum of the series is $\$$ 800.00 million.
D)The geometric series is $\sum _ { n = 0 } ^ { \infty } 200 ( 0.75 ) ^ { n }$ .The sum of the series is $\$$ 15,000 million.
E)The geometric series is $\sum _ { n = 1 } ^ { \infty } 200 ( 25 ) ^ { n + 1 }$ .The sum of the series is $\$$ 150.00 million.

Question 74

You accept a job that pays a salary of $\$$ 50,000 the first year. During the next 39 years, you will receive a 4% raise each year. What would be your total compensation over the 40-year period? Round your answer to the nearest integer.

A) $\$$ 4,751,276
B) $\$$ 1,250,000
C) $\$$ 48,000
D) $\$$ 200,000
E) $\$$ 2,000

Question 75

Consider the sequence (An) whose nth term is given by An $= P \left[ 1 + \frac { r } { 12 } \right] ^ { n }$ where P is the principal, An is the amount of compound interest after n months, and r is the annual percentage rate. Write the first four terms of the sequence for P = $\$$ 8,000 and r = 0.04. Round your answer to two decimal places.

A)8026.67, 8063.49, 8101.43, 8107.20
B)8027.67, 8063.49, 8101.43, 8135.19
C)8026.67, 8053.42, 8080.27, 8107.20
D)8027.67, 8063.49, 8080.27, 8107.20
E)8027.67, 8053.42, 8080.27, 8135.19

Question 76

A factory is polluting a river such that at every mile down river from the factory an environmental expert finds 20% less pollutant than at the preceding mile. If the pollutant's concentration is 700 ppm (parts per million) at the factory, what is its concentration 15 miles down river?

A)140.00 ppm
B)300.00 ppm
C)24.63 ppm
D)875.00 ppm
E)30.79 ppm

Question 77

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( - \frac { 7 } { 8 } \right) ^ { n }$

A) $\frac { 27 } { 5 }$
B) $\frac { 21 } { 5 }$
C) $\frac { 72 } { 13 }$
D) $\frac { 24 } { 5 }$
E) $\frac { 81 } { 13 }$

Question 78

A company produces a new product for which it estimates the annual sales to be 5000 units. Suppose that in any given year $10$ % of the units (regardless of age) will become inoperative. How many units will be in use after n years?

A) $50,000 \left( 1 - 0.9 ^ { n } \right)$
B) $50,000 \left( 0.1 ^ { n } \right)$
C) $5000 \left( 1 - 0.9 ^ { n } \right)$
D) $5000 \left( 0.9 ^ { n } \right)$
E) $50,000 \left( 1 - 0.1 ^ { n } \right)$

Question 79

Give an example of a sequence that converges to $\frac { 1 } { 4 }$ .

A) $a _ { n } = \frac { 8 n ^ { 4 } - 5 } { 2 n ^ { 4 } - 6 }$
B) $a _ { n } = \frac { 2 + 5 n ^ { 4 } } { 8 + 6 n ^ { 4 } }$
C) $a _ { n } = \frac { 3 n ^ { 4 } - 5 } { 9 n ^ { 4 } - 6 }$
D) $a _ { n } = \frac { 2 n ^ { 4 } - 5 } { 8 n ^ { 4 } - 6 }$
E) $a _ { n } = \frac { 3 + 5 n ^ { 4 } } { 9 + 6 n ^ { 4 } }$

Question 80

Bouncing Ball. A ball dropped from a height of 35 feet bounces to $1 / 2$ of its former height with each bounce. Find the total vertical distance that the ball travels.

A)105 feet
B)140 feet
C)64 feet
D)53 feet
E)70 feet