Deck 11: Sequences, Series, and the Binomial Theorem

Question 1

Find $S _ { 1 } , S _ { 2 }$ and $S _ { 6 }$ for $a _ { n } = 8 n + 9$ .

A) $S _ { 1 } = 17 ; S _ { 2 } = 34 ; S _ { 6 } = 102$
B) $S _ { 1 } = 17 ; S _ { 2 } = 25 ; S _ { 6 } = 57$
C) $S _ { 1 } = 17 ; S _ { 2 } = 25 ; S _ { 6 } = 177$
D) $S _ { 1 } = 17 ; S _ { 2 } = 33 ; S _ { 6 } = 177$
E) $S _ { 1 } = 17 ; S _ { 2 } = 42 ; S _ { 6 } = 222$

Answer 1

$S _ { 1 } = 17 ; S _ { 2 } = 42 ; S _ { 6 } = 222$

Question 2

Write the first five terms of the sequence $\left( - \frac { 1 } { 4 } \right) ^ { n + 2 }$ . Assume that n begins with 1.

A) $a _ { 1 } = - \frac { 1 } { 64 } , a _ { 2 } = \frac { 1 } { 256 } , a _ { 3 } = - \frac { 1 } { 1024 } , a _ { 4 } = \frac { 1 } { 4096 } , a _ { 5 } = - \frac { 1 } { 16384 }$
B) $a _ { 1 } = \frac { 1 } { 16 } , a _ { 2 } = \frac { 1 } { 64 } , a _ { 3 } = \frac { 1 } { 256 } , a _ { 4 } = \frac { 1 } { 1024 } , a _ { 5 } = \frac { 1 } { 4096 }$
C) $a _ { 1 } = - \frac { 1 } { 16 } , a _ { 2 } = \frac { 1 } { 64 } , a _ { 3 } = - \frac { 1 } { 256 } , a _ { 4 } = \frac { 1 } { 1024 } , a _ { 5 } = - \frac { 1 } { 4096 }$
D) $a _ { 1 } = \frac { 1 } { 16 } , a _ { 2 } = - \frac { 1 } { 64 } , a _ { 3 } = \frac { 1 } { 256 } , a _ { 4 } = - \frac { 1 } { 1024 } , a _ { 5 } = \frac { 1 } { 4096 }$
E) $a _ { 1 } = \frac { 1 } { 64 } , a _ { 2 } = \frac { 1 } { 256 } , a _ { 3 } = \frac { 1 } { 1024 } , a _ { 4 } = \frac { 1 } { 4096 } , a _ { 5 } = \frac { 1 } { 16384 }$

Answer 2

$a _ { 1 } = - \frac { 1 } { 64 } , a _ { 2 } = \frac { 1 } { 256 } , a _ { 3 } = - \frac { 1 } { 1024 } , a _ { 4 } = \frac { 1 } { 4096 } , a _ { 5 } = - \frac { 1 } { 16384 }$

Question 3

Write an expression for the n th term of the sequence $3,8,13,18,23 , \ldots$ Assume that n begins with 1.

A) $a _ { n } = 3 + 5 n$
B) $a _ { n } = 3 + 6 n$
C) $a _ { n } = 3 + 6 ( n + 1 )$
D) $a _ { n } = 3 + 5 ( n - 1 )$
E) $a _ { n } = 3 + 5 ( n + 1 )$

Answer 3

$a _ { n } = 3 + 5 ( n - 1 )$

Question 4

Write the first five terms of the sequence $a _ { n } = \frac { ( n + 6 ) ! } { ( n + 3 ) ! }$ . Assume that n begins with 1.

A) $a _ { 1 } = 120 , a _ { 2 } = 210 , a _ { 3 } = 336 , a _ { 4 } = 504 , a _ { 5 } = 720$
B) $a _ { 1 } = 202 , a _ { 2 } = 330 , a _ { 3 } = 502 , a _ { 4 } = 719 , a _ { 5 } = 990$
C) $a _ { 1 } = 210 , a _ { 2 } = 336 , a _ { 3 } = 504 , a _ { 4 } = 720 , a _ { 5 } = 990$
D) $a _ { 1 } = 103 , a _ { 2 } = 194 , a _ { 3 } = 328 , a _ { 4 } = 501 , a _ { 5 } = 719$
E) $a _ { 1 } = 336 , a _ { 2 } = 504 , a _ { 3 } = 720 , a _ { 4 } = 990 , a _ { 5 } = 1320$

Question 5

Write the first five terms of the sequence $a _ { n } = \frac { - 3 + \left( - 5 ^ { n } \right) } { n ! }$ . Assume that n begins with 1.

A) $a _ { 1 } = - 2 , a _ { 2 } = 14 , a _ { 3 } = - \frac { 61 } { 3 } , a _ { 4 } = \frac { 157 } { 6 } , a _ { 5 } = - \frac { 1561 } { 60 }$
B) $a _ { 1 } = 8 , a _ { 2 } = - 11 , a _ { 3 } = \frac { 64 } { 3 } , a _ { 4 } = - \frac { 311 } { 12 } , a _ { 5 } = \frac { 391 } { 15 }$
C) $a _ { 1 } = - 2 , a _ { 2 } = 14 , a _ { 3 } = \frac { 61 } { 3 } , a _ { 4 } = - \frac { 311 } { 6 } , a _ { 5 } = - \frac { 391 } { 15 }$
D) $a _ { 1 } = 2 , a _ { 2 } = - 14 , a _ { 3 } = \frac { 61 } { 3 } , a _ { 4 } = - \frac { 157 } { 6 } , a _ { 5 } = \frac { 1561 } { 60 }$
E) $a _ { 1 } = - 8 , a _ { 2 } = 11 , a _ { 3 } = - \frac { 64 } { 3 } , a _ { 4 } = \frac { 311 } { 12 } , a _ { 5 } = - \frac { 391 } { 15 }$

Question 6

Write an expression for the n th term of the sequence $2,5,10,17,26,....$ Assume that n begins with 1.

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

Question 7

Simplify the expression $\frac { 8 ! } { 11 ! }$ .

A) $\frac { 1 } { 729 }$
B) $\frac { 1 } { 990 }$
C) $\frac { 1 } { 720 }$
D) 990
E) 720

Question 8

Write an expression for the n th term of the sequence $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \ldots$ Assume that n begins with 1.

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

Question 9

Write the first five terms of the sequence $( - 1 ) ^ { n } 9 n$ . Assume that n begins with 1.

A) $a _ { 1 } = 0 , a _ { 2 } = - 9 , a _ { 3 } = 18 , a _ { 4 } = - 27 , a _ { 5 } = 36$
B) $a _ { 1 } = 9 , a _ { 2 } = 18 , a _ { 3 } = 27 , a _ { 4 } = 36 , a _ { 5 } = 45$
C) $a _ { 1 } = 0 , a _ { 2 } = - 18 , a _ { 3 } = 27 , a _ { 4 } = - 36 , a _ { 5 } = 45$
D) $a _ { 1 } = - 9 , a _ { 2 } = 18 , a _ { 3 } = - 27 , a _ { 4 } = 36 , a _ { 5 } = - 45$
E) $a _ { 1 } = 9 , a _ { 2 } = - 18 , a _ { 3 } = 27 , a _ { 4 } = - 36 , a _ { 5 } = 45$

Question 10

Write an expression for the n th term of the sequence $\frac { 8 } { 9 } , \frac { 9 } { 10 } , \frac { 10 } { 11 } , \frac { 11 } { 12 } , \frac { 12 } { 13 } ,....$ Assume that n begins with 1.

A) $a _ { n } = \frac { n + 7 } { n + 8 }$
B) $a _ { n } = \frac { n + 8 } { n + 9 }$
C) $a _ { n } = \frac { ( n + 8 ) ! } { ( n + 9 ) ! }$
D) $a _ { n } = \frac { n + 9 } { n + 10 }$
E) $a _ { n } = \frac { ( n + 7 ) ! } { ( n + 8 ) ! }$

Question 11

Write an expression for the n th term of the sequence $4 , \frac { 4 } { 2 } , \frac { 4 } { 6 } , \frac { 4 } { 24 } , \frac { 4 } { 120 } , \ldots$ Assume that n begins with 1.

A) $a _ { n } = 4 + \frac { 1 } { n ! }$
B) $a _ { n } = \frac { 4 } { ( n + 1 ) }$
C) $a _ { n } = \frac { 4 } { n ( n + 1 ) }$
D) $a _ { n } = \frac { 4 } { n ! }$
E) $a _ { n } = 4 + \frac { 1 } { ( n + 1 ) ! }$

Question 12

Write the first five terms of the sequence $a _ { n } = \frac { n } { 9 n + 1 }$ . Assume that n begins with 1.

A) $a _ { 1 } = - \frac { 1 } { 10 } , a _ { 2 } = - \frac { 2 } { 19 } , a _ { 3 } = - \frac { 3 } { 28 } , a _ { 4 } = - \frac { 4 } { 37 } , a _ { 5 } = - \frac { 5 } { 46 }$
B) $a _ { 1 } = \frac { 1 } { 10 } , a _ { 2 } = \frac { 2 } { 19 } , a _ { 3 } = \frac { 3 } { 28 } , a _ { 4 } = \frac { 4 } { 37 } , a _ { 5 } = \frac { 5 } { 46 }$
C) $a _ { 1 } = 0 , a _ { 2 } = - \frac { 1 } { 10 } , a _ { 3 } = \frac { 2 } { 19 } , a _ { 4 } = - \frac { 3 } { 28 } , a _ { 5 } = \frac { 4 } { 37 }$
D) $a _ { 1 } = - \frac { 1 } { 10 } , a _ { 2 } = \frac { 2 } { 19 } , a _ { 3 } = - \frac { 3 } { 28 } , a _ { 4 } = \frac { 4 } { 37 } , a _ { 5 } = - \frac { 5 } { 46 }$
E) $a _ { 1 } = 0 , a _ { 2 } = \frac { 1 } { 10 } , a _ { 3 } = \frac { 2 } { 19 } , a _ { 4 } = \frac { 3 } { 28 } , a _ { 5 } = \frac { 4 } { 37 }$

Question 13

Write an expression for the n th term of the sequence $1,3 , \frac { 3 ^ { 2 } } { 2 } , \frac { 3 ^ { 3 } } { 6 } , \frac { 3 ^ { 4 } } { 24 } , \frac { 3 ^ { 5 } } { 120 } , \ldots$ Assume that n begins with 1.

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

Question 14

Find the partial sum $\sum _ { i = 0 } ^ { 5 } 6 i + 2$ .

A) 104
B) 140
C) 102
D) 142
E) 120

Question 15

Find $a _ { 8 }$ of the sequence $a _ { n } = \frac { n ^ { 2 } - 7 } { ( n - 2 ) ! }$ .

A) $a _ { 8 } = \frac { 32 } { 361 }$
B) $a _ { 8 } = - \frac { 32 } { 361 }$
C) $a _ { 8 } = - \frac { 19 } { 1680 }$
D) $a _ { 8 } = \frac { 19 } { 1680 }$
E) $a _ { 8 } = \frac { 19 } { 240 }$

Question 16

Write the first five terms of the sequence $a _ { n } = \frac { ( - 1 ) ^ { n + 4 } } { n ^ { 2 } + 1 }$ . Assume that n begins with 1.

A) $a _ { 1 } = - \frac { 1 } { 2 } , a _ { 2 } = \frac { 1 } { 5 } , a _ { 3 } = - \frac { 1 } { 10 } , a _ { 4 } = \frac { 1 } { 17 } , a _ { 5 } = - \frac { 1 } { 26 }$
B) $a _ { 1 } = \frac { 1 } { 2 } , a _ { 2 } = \frac { 1 } { 5 } , a _ { 3 } = \frac { 1 } { 10 } , a _ { 4 } = \frac { 1 } { 17 } , a _ { 5 } = \frac { 1 } { 26 }$
C) $a _ { 1 } = - 1 , a _ { 2 } = \frac { 1 } { 4 } , a _ { 3 } = - \frac { 1 } { 9 } , a _ { 4 } = \frac { 1 } { 16 } , a _ { 5 } = - \frac { 1 } { 25 }$
D) $a _ { 1 } = \frac { 1 } { 2 } , a _ { 2 } = - \frac { 1 } { 5 } , a _ { 3 } = \frac { 1 } { 10 } , a _ { 4 } = - \frac { 1 } { 17 } , a _ { 5 } = \frac { 1 } { 26 }$
E) $a _ { 1 } = 1 , a _ { 2 } = - \frac { 1 } { 4 } , a _ { 3 } = \frac { 1 } { 9 } , a _ { 4 } = - \frac { 1 } { 16 } , a _ { 5 } = \frac { 1 } { 25 }$

Question 17

Write an expression for the n th term of the sequence $7 + \frac { 2 } { 1 } , 7 + \frac { 2 } { 2 } , 7 + \frac { 2 } { 3 } , 7 + \frac { 2 } { 4 } , 7 + \frac { 2 } { 5 } , \cdots$ Assume that n begins with 1.

A) $a _ { n } = \frac { 8 n + 3 } { n }$
B) $a _ { n } = \frac { 3 n + 8 } { n }$
C) $a _ { n } = \frac { 7 n + 2 } { n }$
D) $a _ { n } = \frac { 2 n + 7 } { n }$
E) $a _ { n } = \frac { 9 } { n }$

Question 18

Match the sequence $a _ { n } = \frac { 20 } { n + 1 }$ with the graph of its first 10 terms.

A)
B)
C)
D)
E)

Question 19

Write the first five terms of the sequence $a _ { n } = - 4 - \frac { 1 } { 3 ^ { n } }$ . Assume that n begins with 1.

A) $a _ { 1 } = - \frac { 11 } { 3 } , a _ { 2 } = - \frac { 35 } { 9 } , a _ { 3 } = - \frac { 107 } { 27 } , a _ { 4 } = - \frac { 323 } { 81 } , a _ { 5 } = - \frac { 971 } { 243 }$
B) $a _ { 1 } = - \frac { 13 } { 3 } , a _ { 2 } = - \frac { 37 } { 9 } , a _ { 3 } = - \frac { 109 } { 27 } , a _ { 4 } = - \frac { 325 } { 81 } , a _ { 5 } = - \frac { 973 } { 243 }$
C) $a _ { 1 } = \frac { 11 } { 3 } , a _ { 2 } = \frac { 35 } { 9 } , a _ { 3 } = \frac { 107 } { 27 } , a _ { 4 } = \frac { 323 } { 81 } , a _ { 5 } = \frac { 971 } { 243 }$
D) $a _ { 1 } = - 3 , a _ { 2 } = \frac { 11 } { 3 } , a _ { 3 } = \frac { 35 } { 9 } , a _ { 4 } = \frac { 107 } { 27 } , a _ { 5 } = \frac { 323 } { 81 }$
E) $a _ { 1 } = - 5 , a _ { 2 } = - \frac { 13 } { 3 } , a _ { 3 } = - \frac { 37 } { 9 } , a _ { 4 } = - \frac { 109 } { 27 } , a _ { 5 } = - \frac { 325 } { 81 }$

Question 20

Write the first five terms of the sequence $a _ { n } = - \frac { 3 } { n + 4 }$ . Assume that n begins with 1.

A) $a _ { 1 } = \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = \frac { 1 } { 3 }$
B) $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = - \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$
C) $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$
D) $a _ { 1 } = - \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = - \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = - \frac { 3 } { 8 }$
E) $a _ { 1 } = - \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = - \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = - \frac { 1 } { 3 }$

Question 21

Find a formula for the n th term of the arithmetic sequence. $a _ { 5 } = 22 , a _ { 6 } = 29$

A) $7 n - 13$
B) $7 n - 20$
C) $7 n + 15$
D) $7 n + 29$
E) $7 n - 6$

Question 22

Find the common difference of the arithmetic sequence $34,45,56,67 , \ldots$

A) 22
B) 11
C) -22
D) 34
E) -11

Question 23

Write the sum using sigma notation. Begin with $k = 1$ . $\frac { 1 } { 1 ^ { 3 } } + \frac { 1 } { 2 ^ { 3 } } + \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 4 ^ { 3 } } + \ldots + \frac { 1 } { 16 ^ { 3 } }$

A) $\sum _ { k = 1 } ^ { 15 } \frac { 1 } { k ^ { 3 } }$
B) $\sum _ { k = 1 } ^ { 15 } \frac { 1 } { ( k - 1 ) ^ { 3 } }$
C) $\sum _ { k = 1 } ^ { 16 } \frac { 1 } { ( k - 1 ) ^ { 3 } }$
D) $\sum _ { k = 1 } ^ { 16 } \frac { 1 } { ( k + 1 ) ^ { 3 } }$
E) $\sum _ { k = 1 } ^ { 16 } \frac { 1 } { k ^ { 3 } }$

Question 24

Write the sum using sigma notation. Begin with $k = 0$ . $23 + 26 + 29 + 32 + 35$

A) $\sum _ { k = 0 } ^ { 4 } 3 k + 23$
B) $\sum _ { k = 0 } ^ { 5 } 3 k + 23$
C) $\sum _ { k = 0 } ^ { 5 } 23 k + 20$
D) $\sum _ { k = 0 } ^ { 4 } 23 k + 20$
E) $\sum _ { k = 0 } ^ { 4 } 23 k + 3$

Question 25

Find the partial sum of $\sum _ { k = 1 } ^ { 30 } ( k + 8 )$ .

A) 930
B) 705
C) 255
D) 240
E) 2280

Question 26

Find a formula for the n th term of the arithmetic sequence. $a _ { 29 } = 16 , d = \frac { 3 } { 7 }$

A) $a _ { n } = \frac { 3 n + 31 } { 7 }$
B) $a _ { n } = \frac { 3 n + 4 } { 7 }$
C) $a _ { n } = \frac { 3 n + 1 } { 7 }$
D) $a _ { n } = \frac { 3 n + 28 } { 7 }$
E) $a _ { n } = \frac { 3 n + 25 } { 7 }$

Question 27

Find a formula for the n th term of the arithmetic sequence. $a _ { 1 } = 11 , d = 4$

A) $\alpha _ { n } = 4 n + 11$
B) $a _ { n } = 4 n + 15$
C) $a _ { n } = 4 n + 7$
D) $a _ { n } = \frac { n } { 2 } ( 11 + 4 n )$
E) $a _ { n } = \frac { n } { 2 } ( 11 + 4 ( n - 1 ) )$

Question 28

Write the first five terms of the arithmetic sequence defined recursively. $a _ { 1 } = 50 , a _ { k + 1 } = a _ { k } - 5$

A) $a _ { 1 } = 70 , a _ { 2 } = 65 , a _ { 3 } = 60 , a _ { 4 } = 55 , a _ { 5 } = 50$
B) $a _ { 1 } = 50 , a _ { 2 } = 55 , a _ { 3 } = 60 , a _ { 4 } = 65 , a _ { 5 } = 70$
C) $a _ { 1 } = 50 , a _ { 2 } = 45 , a _ { 3 } = 40 , a _ { 4 } = 35 , a _ { 5 } = 30$
D) $a _ { 1 } = 30 , a _ { 2 } = 35 , a _ { 3 } = 40 , a _ { 4 } = 45 , a _ { 5 } = 50$
E) $a _ { 1 } = 50 , a _ { 2 } = 40 , a _ { 3 } = 45 , a _ { 4 } = 30 , a _ { 5 } = 35$

Question 29

Find a formula for the n th term of the arithmetic sequence. $a _ { 5 } = 13 , a _ { 9 } = 3$

A) $a _ { n } = - \frac { 5 n - 51 } { 2 }$
B) $a _ { n } = \frac { 5 n - 51 } { 2 }$
C) $a _ { n } = 23 n - 10$
D) $\alpha _ { n } = 13 n + 10$
E) $a _ { n } = - \frac { 5 n + 51 } { 2 }$

Question 30

Find the partial sum $\sum _ { n = 0 } ^ { 5 } \left( - \frac { 1 } { 5 } \right) ^ { n }$

A) $\frac { 521 } { 3125 }$
B) $\frac { 2604 } { 3125 }$
C) $- \frac { 104 } { 3125 }$
D) $- \frac { 2604 } { 3125 }$
E) $\frac { 521 } { 3125 }$ .

Question 31

Determine whether the sequence $\ln 7 , \ln 14 , \ln 21 , \ln 28 , \ldots$ is arithmetic. If so, find the common difference.

A)arithmetic; 14
B)arithmetic; 2
C)arithmetic;7
D)arithmetic; 28
E)not arithmetic

Question 32

Write the sum using sigma notation. Begin with $k = 0$ . $\frac { 1 } { 9 ^ { 0 } } - \frac { 1 } { 9 ^ { 1 } } + \frac { 1 } { 9 ^ { 2 } } - \frac { 1 } { 9 ^ { 3 } } + \ldots - \frac { 1 } { 9 ^ { 11 } }$

A) $\sum _ { k = 0 } ^ { 10 } - \frac { 1 } { 9 ^ { k } }$
B) $\sum _ { k = 0 } ^ { 11 } \left( - \frac { 1 } { 9 } \right) ^ { k + 1 }$
C) $\sum _ { k = 0 } ^ { 10 } \left( - \frac { 1 } { 9 } \right) ^ { k + 1 }$
D) $\sum _ { k = 0 } ^ { 11 } - \frac { 1 } { 9 ^ { k } }$
E) $\sum _ { k = 0 } ^ { 11 } \left( - \frac { 1 } { 9 } \right) ^ { k }$

Question 33

Determine whether the sequence is arithmetic. If so, find the common difference. $\frac { 1 } { 5 } , \frac { 3 } { 5 } , \frac { 9 } { 5 } , \frac { 27 } { 5 } , \frac { 81 } { 5 } , \ldots$

A) $\frac { 3 } { 5 }$
B) $\frac { 9 } { 5 }$
C) 2
5
D) $\frac { 81 } { 5 }$
E)The sequence is not arithmetic.

Question 34

Find the partial sum $\sum _ { j = 1 } ^ { 3 } \frac { ( - 1 ) ^ { j } } { j ^ { 3 } }$ .

A) $- \frac { 235 } { 216 }$
B) $\frac { 65 } { 216 }$
C) $- \frac { 65 } { 216 }$
D) $- \frac { 197 } { 216 }$
E) $\frac { 197 } { 216 }$

Question 35

Find a formula for the n th term of the arithmetic sequence. $a _ { 1 } = 75 , d = - 8$

A) $a _ { n } = - 8 n - 83$
B) $a _ { n } = - 8 n + 83$
C) $a _ { n } = 8 n + 75$
D) $a _ { n } = - 8 n + 67$
E) $a _ { n } = - 8 n + 75$

Question 36

A deposit of $\$ 3,000$ is made in an account that earns $4 \%$ interest compounded yearly. The balance in the account after N years is given by $A _ { N } = 3,000 ( 1 + 0.04 ) ^ { N }$ , $N = 1,2,3 , \ldots$ Find the balance in this account after $20$ years by computing $A _ { 20 }$ . Round your answer to the nearest cent.

A) $A _ { 20 } = \$ 5,403$
B) $A _ { 20 } = \$ 62,400$
C) $A _ { 20 } = \$ 6,573$
D) $A _ { 20 } = \$ 3,245$
E) $A _ { 20 } = \$ 6,321$

Question 37

Write the first five terms of the arithmetic sequence defined recursively. $a _ { 1 } = 12 , a _ { k + 1 } = a _ { k } + 6$

A) $a _ { 1 } = 18 , a _ { 2 } = 30 , a _ { 3 } = 42 , a _ { 4 } = 54 , a _ { 5 } = 66$
B) $a _ { 1 } = 12 , a _ { 2 } = 24 , a _ { 3 } = 36 , a _ { 4 } = 48 , a _ { 5 } = 60$
C) $a _ { 1 } = 12 , a _ { 2 } = 24 , a _ { 3 } = 18 , a _ { 4 } = 36 , a _ { 5 } = 30$
D) $a _ { 1 } = 18 , a _ { 2 } = 24 , a _ { 3 } = 30 , a _ { 4 } = 36 , a _ { 5 } = 42$
E) $a _ { 1 } = 12 , a _ { 2 } = 18 , a _ { 3 } = 24 , a _ { 4 } = 30 , a _ { 5 } = 36$

Question 38

Find the common difference of the arithmetic sequence with the n th term $a _ { n } = 5 - 6 n$ .

A) -5
B) -6
C) 6
D) -1
E) 5

Question 39

Find the common difference of the arithmetic sequence $\frac { 17 } { 8 } , \frac { 7 } { 4 } , \frac { 11 } { 8 } , 1 , \frac { 5 } { 8 } , \ldots$

A) $\frac { 11 } { 4 }$
B) $- \frac { 3 } { 8 }$
C) $- \frac { 3 } { 4 }$
D) $- \frac { 7 } { 4 }$
E) $\frac { 11 } { 8 }$

Question 40

Determine whether the sequence is arithmetic. If so, find the common difference. $\ln 4 , \ln 8 , \ln 12 , \ln 16 , \ln 20 , \ldots$

A) 8
B) 4
C) $\ln 8$
D) $\ln 4$
E)The sequence is not arithmetic.

Question 41

Find the common ratio of the geometric sequence. $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16},...$

A) $-\frac{1}{8}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $-\frac{1}{2}$
E) $-\frac{1}{4}$

Question 42

Find the common ratio of the geometric sequence. $\frac{1}{8 \pi}, \frac{1}{(8 \pi)^{2}}, \frac{1}{(8 \pi)^{3}}, \frac{1}{(8 \pi)^{4}},....$

A) $\frac{1}{\pi^{2}}$
B) $\frac{1}{\pi}$
C) $\frac{1}{8 \pi}$
D) $\frac{1}{8}$
E) $\frac { 1 } { ( 8 \pi ) ^ { 2 } }$

Question 43

Write the first five terms of the geometric sequence, given $a_{1}=-9$ and $r=-\frac{1}{3}$ .

A) $-9,3,-1, \frac{1}{3},-\frac{1}{9}$
B) $-9,-3,1,-\frac{1}{3},-\frac{1}{9}$
C) $-9,27,-1, \frac{1}{9},-\frac{1}{27}$
D) $-27,9,1, \frac{1}{9},-\frac{1}{27}$
E) $- 27,9 , - 1 , - \frac { 1 } { 3 } , - \frac { 1 } { 27 }$

Question 44

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $a_{1}=3, r=\frac{7}{4}$

A) $a_{n}=3\left(\frac{7}{4}\right)^{n-1}$
B) $a_{n}=\left(\frac{21}{4}\right)^{n-1}$
C) $a_{n}=\left(\frac{7}{4}\right)^{n}$
D) $a_{n}=\left(\frac{21}{4}\right)^{n}$
E) $a_{n}=3\left(\frac{7}{4}\right)^{n}$

Question 45

Find the sum of the first positive integers.

A)
B)
C)
D)
E)

Question 46

Find the sum of the first positive even integers.

A)
B)
C)
D)
E)

Question 47

A jogger decides to take a jog of $12$ miles. The first two miles take the jogger $8$ minutes and $10$ minutes respectively to complete each mile. The jogger estimates that this pattern will continue for $12$ miles. Estimate the time required to jog $12$ miles.

A) $240$ minutes
B) $228$ minutes
C) $180$ minutes
D) $168$ minutes
E) $264$ minutes

Question 48

Determine whether the sequence is geometric. If so, find the common ratio. $9,15,21,27, \ldots$

A)9
B)15
C) $\frac{5}{2}$
D)6
E)not geometric

Question 49

Determine whether the sequence is geometric. If so, find the common ratio. $1024,256,64,16, \ldots$

A)-4
B)4
C) $-\frac{1}{4}$
D) $\frac{1}{4}$
E)not geometric

Question 50

Find the common ratio of the geometric sequence. $-11,11,-11,11,...$

A) 1
B)-11
C) -1
D) $-\frac{1}{11}$
E)11

Question 51

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $a_{1}=10, a_{2}=2$

A) $a_{n}=10\left(\frac{1}{5}\right)^{n-1}$
B) $a_{n}=2^{n-1}$
C) $a_{n}=2(5)^{n-1}$
D) $a_{n}=2(5)^{n}$
E) $a_{n}=10\left(\frac{1}{5}\right)^{x}$

Question 52

Find the n th partial sum of the arithmetic sequence. $8,13,18,23,28 , \ldots , n = 12$

A) 390
B) 438
C) 426
D) 354
E) 402

Question 53

Find the partial sum of $\sum _ { k = 1 } ^ { 5 } \frac { n } { 10 }$ .

A) -13.75
B) 0.25
C) 2.5
D) 1.5
E) 3

Question 54

Find the n th partial sum of the arithmetic sequence. $9,16,23,30,37 , \ldots , n = 10$

A) 440
B) 790
C) 405
D) 395
E) 880

Question 55

Determine whether the sequence is geometric. If so, find the common ratio. $1,-\frac{7}{2}, \frac{49}{4},-\frac{343}{8}, \frac{2401}{16},..$

A) $\frac{2}{7}$
B) - $\frac{2}{7}$
C) $\frac{7}{2}$
D) - $\frac{7}{2}$
E)not geometric

Question 56

A museum only allows patrons to enter at fifteen minute intervals. An observer notices that every weekday at 2:00 there are $8$ people waiting at the entrance and at 3:00 there are $20$ people waiting. In addition, there are $10$ people waiting at the entrance at all 15-minute intervals that are not on the hour. This pattern continues through 6:00. How many people wait to enter the museum in the interval from 2:00 to 6:00?

A) $280$
B) $232$
C) $291$
D) $248$
E) $208$

Question 57

In section 112 of Hammer Stadium there are $25$ rows of seats: $16$ in the first row, $17$ seats in the second row, $18$ seats in the third row, and so on. What is the cost of each ticket if the total cost of all the seats in section 112 is $\$ 29,400$ ? Round the answer to two decimal places.

A) $\$ 42.00$
B) $\$ 43.25$
C) $\$ 48.75$
D) $\$ 45.75$
E) $\$ 39.50$

Question 58

Find the n th partial sum of the arithmetic sequence. $a _ { 1 } = 0.8 , a _ { 5 } = 2.8 , \ldots , n = 13$

A) 52.65
B) 92.30
C) 105.30
D) 46.15
E) 49.40

Question 59

In your new job as an accountant you are told that your starting salary will be $\$ 36,000$ with an increase of $\$ 2500$ at the end of each of the first $4$ years. How much will you be paid through the end of your first 5 years of employment with the company?

A)$ $211,250$
B)$ $164,000$
C)$ $159,000$
D)$ $205,000$
E)$ $198,750$

Question 60

Find the n th partial sum of the arithmetic sequence. $300,257,214,171,128 \ldots , n = 9$

A) 1926
B) 2304
C) -198
D) 958.5
E) 1152

Question 61

Find the n th partial sum of the geometric sequence. $5,15,45,135,405, \ldots, \quad n=7$

A)1,820
B)5,465
C)200
D)5,514
E)5,808

Question 62

Find the balance of an increasing annuity in which a principal of $300 is invested each month for 33 years, compounded monthly at a rate of 8%.

A)$42,454.33
B)$583,960.90
C)$63,895.28
D)$123,835.77
E)$11,406.08

Question 63

Find the partial sum. $\sum_{i=1}^{4} 2\left(\frac{3}{2}\right)^{i-1}$

A) $\frac{105}{4}$
B) $\frac{17}{4}$
C) $\frac{65}{4}$
D) $\frac{89}{4}$
E) $\frac{79}{4}$

Question 64

Evaluate the binomial coefficient $${ } _ { 15 } C _ { 13 }$$ .

A) $120$
B) $300$
C) $118$
D) $317$
E) $105$

Question 65

Find the partial sum. $\sum_{i-1}^{8} 3\left(-\frac{1}{4}\right)^{i-1}$

A) $\frac{9}{16,384}$
B) $\frac{291}{16,384}$
C) $\frac{39,321}{16,384}$
D) $\frac{3}{16,384}$
E) $\frac{99}{16,384}$

Question 66

A ball is dropped from a height of 10 feet. Each time it drops h feet, it rebounds 0.79 h feet. Find the total distance traveled by the ball. Round your answer to two decimal places.

A)75.24
B)37.62
C)47.62
D)85.24
E)22.66

Question 67

Find the n th partial sum of the geometric sequence. Round your answer to 2 decimal places. $22,22(1.04), 22(1.04)^{2}, 22(1.04)^{3}, \ldots, \quad n=9$

A)330.57
B)173.76
C)752.71
D)232.82
E)296.70

Question 68

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $4,-10,25,-\frac{125}{2}, \ldots$

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

Question 69

You accept a job as a manager that pays a salary of $34,000 the first year. During the next 39 years, you receive a 4% raise each year. What would your total salary be over the 40 year period? Round answer to the nearest cent.

A)$3,740,420.98
B)$3,230,867.53
C)$3,394,102.24
D)$2,922,991.43
E)$2,638,342.67

Question 70

Find the sum. $\sum_{n=0}^{\infty}\left(\frac{1}{4}\right)^{n}$

A) $\frac{4}{5}$
B) $\frac{5}{4}$
C) $\frac{1}{5}$
D) $\frac{4}{3}$
E) $\frac{1}{6}$

Question 71

A company buys a machine for $400,000. During the next 8 years, the machine depreciates at the rate of 25% per year. That is, at the end of each year, the depreciated value is 75% of what it was at the beginning of the year. Find the depreciated value of the machine at the end of the 8 full years. Round answer to the nearest cent.

A)24.41
B)30,033.87
C)6.10
D)53,393.55
E)40,045.17

Question 72

Find the balance of an increasing annuity in which a principal of $20 is invested each month for 40 years, compounded monthly at a rate of 8%.

A)$939.44
B)$11,878.94
C)$5,744.14
D)$70,305.62
E)$3,703.31

Question 73

Find the n th partial sum of the geometric sequence. $1,-4,16,-64,256, \ldots, \quad n=7$

A)3326
B)3277
C)-819
D)3620
E)-51

Question 74

Find the n th partial sum of the geometric sequence. $-64,16,-4,1,-\frac{1}{4}, \ldots, \quad n=6$

A) $-\frac{819}{16}$
B) $-\frac{3}{16}$
C) $\frac{1}{16}$
D) $\frac{3277}{16}$
E) $-\frac{51}{16}$

Question 75

A city of 700,000 people is growing at the rate of 2% per year. That is, at the end of each year, the population is 1.02 times the population at the beginning of the year. Estimate the population years 19 from now. Round to the nearest integer.

A)1,243,091
B)1,019,768
C)1,103,829
D)1,082,186
E)1,194,821

Question 76

Find the sum. $\sum_{n=0}^{\infty}\left(-\frac{3}{8}\right)^{n}$

A) $\frac{8}{11}$
B) $\frac{3}{14}$
C) $\frac{5}{8}$
D) $\frac{8}{5}$
E) $\frac{3}{5}$

Question 77

Evaluate the binomial coefficient $${ } _ { 20 } C _ { 20 }$$ .

A)1
B) $0$
C) $19$
D) $20$
E) $2$

Question 78

Find $a_{5}$ of the geometric sequence, given $a_{1}=9$ and $a_{2}=8$

A) $\frac{6561}{4096}$
B) $\frac{4096}{6561}$
C) $\frac{729}{512}$
D) $\frac { 4096 } { 729 }$
E) $\frac { 6561 } { 512 }$

Question 79

Evaluate the binomial coefficient $${ } _ { 7 } C _ { 3 }$$ .

A) $6$
B) $5,034$
C) $840$
D) $35$
E) $210$

Question 80

Find the sum. $21+9+\frac{27}{7}+\frac{81}{49}+\ldots$

A) $\frac{441}{10}$
B) $\frac{147}{4}$
C) $\frac{21}{4}$
D) $\frac{1323}{4}$
E) $\frac{3}{4}$