Question 1

Use mathematical induction to prove the following.
-$$2 ^ { n } > n$$

Accepted Answer

Answers may vary. One possibility: \[\be

Question 2

Find the first term and the common difference.
-$6,9,12,15 , \ldots$

Accepted Answer

A)  $a _ { 1 } = 6 , d = 15$ 
B)  $a _ { 1 } = 6 , d = 3$ 
C)  $a _ { 1 } = 6 , d = 9$ 
D)  $a _ { 1 } = 6 , d = - 3$ 
A)  $a _ { 1 } = 6 , d = 15$ 
B)  $a _ { 1 } = 6 , d = 3$ 
C)  $a _ { 1 } = 6 , d = 9$ 
D)  $a _ { 1 } = 6 , d = - 3$

Question 3

Determine the number of subsets of the set.
-A set of 11 elements

Accepted Answer

A) 2048 
B) 22 
C) 1024 
D) 23 
A) 2048 
B) 22 
C) 1024 
D) 23

Question 4

Solve.
-Find the sum of the first 114 positive multiples of 3.

Accepted Answer

A) 19,323 
B) 19,665 
C) 19,494 
D) 19,838 
A) 19,323 
B) 19,665 
C) 19,494 
D) 19,838

Question 5

Find the indicated quantity.
-a5, when a10 = 37 and a15 = 57

Accepted Answer

A) 4 
B) 1 
C) 17 
D) 21 
A) 4 
B) 1 
C) 17 
D) 21

Question 6

Find fraction notation.
-$0.0 \overline { 1 }$

Accepted Answer

A)  0 
B)  $\frac { 1 } { 50 }$ 
C)  $54 \frac { 1 } { 9 }$ 
D)  $\frac { 1 } { 90 }$ 
A)  0 
B)  $\frac { 1 } { 50 }$ 
C)  $54 \frac { 1 } { 9 }$ 
D)  $\frac { 1 } { 90 }$

Question 7

Solve the problem.
-During a five-year period, a company doubles its profits each year. If the profits at the end of the fifth year are $224,000, then what are the profits for each of the first four years?

Accepted Answer

A) $14,000, $28,000, $42,000, $56,000 
B) $14,000, $28,000, $56,000, $112,000 
C) $15,000, $30,000, $60,000, $118,000 
D) $14,000, $28,000, $56,000, $140,000 
A) $14,000, $28,000, $42,000, $56,000 
B) $14,000, $28,000, $56,000, $112,000 
C) $15,000, $30,000, $60,000, $118,000 
D) $14,000, $28,000, $56,000, $140,000

Question 8

Find the indicated term of the sequence.
-$$a _ { n } = ( 5 n - 7 ) ( 6 n - 5 ) ; a _ { 12 }$$

Accepted Answer

A)  5159 
B)  3551 
C)  4081 
D)  4489 
A)  5159 
B)  3551 
C)  4081 
D)  4489

Question 9

Find the common ratio.
-64, 32, 16, 8, 4, . . .

Accepted Answer

A) 4 
B) -0.5 
C) 2 
D) 0.5 
A) 4 
B) -0.5 
C) 2 
D) 0.5

Question 10

Solve.
-A ball is dropped from a height of $4 \mathrm {~m}$. On each upward bounce the ball returns to $\frac { 3 } { 4 }$ of its previous height. Find the total distance the ball travels in an upward direction before coming to rest.

Accepted Answer

A)  $28 \mathrm {~m}$ 
B)  $12 \mathrm {~m}$ 
C)  $15 \mathrm {~m}$ 
D)  $3 \mathrm {~m}$ 
A)  $28 \mathrm {~m}$ 
B)  $12 \mathrm {~m}$ 
C)  $15 \mathrm {~m}$ 
D)  $3 \mathrm {~m}$

Question 11

Solve.
-In how many distinguishable ways can the letters of the word WINDOW be arranged?

Accepted Answer

A) 120 
B) 720 
C) 360 
D) 36 
A) 120 
B) 720 
C) 360 
D) 36

Question 12

Find the indicated term of the binomial expansion.
-7 th term; $( x + 2 y ) ^ { 10 }$

Accepted Answer

A)  $6720 x ^ { 6 } y ^ { 4 }$ 
B)  $13,440 x ^ { 4 } y ^ { 6 }$ 
C)  $13,440 x ^ { 6 } y ^ { 4 }$ 
D)  $6720 x ^ { 4 } y^7$ 
A)  $6720 x ^ { 6 } y ^ { 4 }$ 
B)  $13,440 x ^ { 4 } y ^ { 6 }$ 
C)  $13,440 x ^ { 6 } y ^ { 4 }$ 
D)  $6720 x ^ { 4 } y^7$

Question 13

Expand.
-$$( x - \sqrt { 7 } ) ^ { 5 }$$

Accepted Answer

A)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 49 x - 49 \sqrt { 7 }$ 
B)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49$ 
C)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$ 
D)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 7 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$ 
A)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 49 x - 49 \sqrt { 7 }$ 
B)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49$ 
C)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 70 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$ 
D)  $x ^ { 5 } - 5 x ^ { 4 } \sqrt { 7 } + 7 x ^ { 3 } - 70 x ^ { 2 } \sqrt { 7 } + 245 x - 49 \sqrt { 7 }$

Question 14

Find the nth, or general, term.
-1, 7, 49, . . .

Accepted Answer

A)  $7 ^ { n - 1 }$ 
B)  $7 ^ { 1 - n }$ 
C)  $7 ^ { \mathrm { n } }$ 
D)  $7 ( \mathrm { n } - 1 )$ 
A)  $7 ^ { n - 1 }$ 
B)  $7 ^ { 1 - n }$ 
C)  $7 ^ { \mathrm { n } }$ 
D)  $7 ( \mathrm { n } - 1 )$

Question 15

Expand.
-$$\left( - 3 x ^ { 2 } - y \right) ^ { 5 }$$

Accepted Answer

A)  $- 243 x ^ { 10 } - 405 x ^ { 8 } y - 270 x ^ { 6 } y ^ { 2 } - 90 x ^ { 4 } y ^ { 3 } - 15 x ^ { 2 } y ^ { 4 } - y ^ { 5 }$ 
B)  $- 243 x + 405 x ^ { 8 } y + 270 x ^ { 6 } y ^ { 2 } + 90 x ^ { 4 } y ^ { 3 } + 15 x ^ { 2 } y ^ { 4 } + y ^ { 5 }$ 
C)  $- 243 x 10 + 405 x ^ { 8 } y - 270 x ^ { 6 } y ^ { 2 } + 90 x ^ { 4 } y ^ { 3 } - 15 x ^ { 2 } y ^ { 4 } + y ^ { 5 }$ 
D)  $- 243 x ^ { 10 } - 405 x ^ { 8 } y + 270 x ^ { 6 } y ^ { 2 } - 90 x ^ { 4 } y ^ { 3 } + 15 x ^ { 2 } y ^ { 4 } - y ^ { 5 }$ 
A)  $- 243 x ^ { 10 } - 405 x ^ { 8 } y - 270 x ^ { 6 } y ^ { 2 } - 90 x ^ { 4 } y ^ { 3 } - 15 x ^ { 2 } y ^ { 4 } - y ^ { 5 }$ 
B)  $- 243 x + 405 x ^ { 8 } y + 270 x ^ { 6 } y ^ { 2 } + 90 x ^ { 4 } y ^ { 3 } + 15 x ^ { 2 } y ^ { 4 } + y ^ { 5 }$ 
C)  $- 243 x 10 + 405 x ^ { 8 } y - 270 x ^ { 6 } y ^ { 2 } + 90 x ^ { 4 } y ^ { 3 } - 15 x ^ { 2 } y ^ { 4 } + y ^ { 5 }$ 
D)  $- 243 x ^ { 10 } - 405 x ^ { 8 } y + 270 x ^ { 6 } y ^ { 2 } - 90 x ^ { 4 } y ^ { 3 } + 15 x ^ { 2 } y ^ { 4 } - y ^ { 5 }$

Question 16

Find the sum.
-$$\sum _ { k = 1 } ^ { 4 } ( 3 k - 2 )$$

Accepted Answer

A)  22 
B)  10 
C)  13 
D)  21 
A)  22 
B)  10 
C)  13 
D)  21

Question 17

Expand.
-$$\left( \frac { 1 } { 3 } x + 2 \right) ^ { 3 }$$

Accepted Answer

A)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 8$ 
B)  $\frac { 1 } { 27 } x ^ { 6 } + \frac { 4 } { 9 } x ^ { 3 } + 8$ 
C)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 4 x + 8$ 
D)  $\frac { 1 } { 27 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 4 x + 8$ 
A)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 8$ 
B)  $\frac { 1 } { 27 } x ^ { 6 } + \frac { 4 } { 9 } x ^ { 3 } + 8$ 
C)  $\frac { 1 } { 9 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 4 x + 8$ 
D)  $\frac { 1 } { 27 } x ^ { 3 } + \frac { 2 } { 3 } x ^ { 2 } + 4 x + 8$

Question 18

Find the indicated term of the sequence.
-$$a _ { n } = 5 n ^ { 2 } ( 15 n - 118 ) ; a _ { 6 }$$

Accepted Answer

A)  $- 6120$ 
B)  $- 5040$ 
C)  38,520 
D)  37,440 
A)  $- 6120$ 
B)  $- 5040$ 
C)  38,520 
D)  37,440

Question 19

Use mathematical induction to prove the following.
-$$1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

Accepted Answer

\[\begin{array} { l } 
1 ^ { 2 } + 4 ^ {

Question 20

Find the first term and the common difference.
-$$\frac { 7 } { 12 } , \frac { 1 } { 3 } , \frac { 1 } { 12 } , - \frac { 1 } { 6 } , \ldots$$

Accepted Answer

A)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 1 } { 4 }$ 
B)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 1 } { 4 }$ 
C)  $a _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 7 } { 12 }$ 
D)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 3 } { 4 }$ 
A)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 1 } { 4 }$ 
B)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 1 } { 4 }$ 
C)  $a _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = \frac { 7 } { 12 }$ 
D)  $\mathrm { a } _ { 1 } = \frac { 7 } { 12 } , \mathrm {~d} = - \frac { 3 } { 4 }$

Use mathematical induction to prove the following. - $2 ^ { n } > n$

Find the first term and the common difference. - $6,9,12,15 , \ldots$

Determine the number of subsets of the set. -A set of 11 elements

Solve. -Find the sum of the first 114 positive multiples of 3.

Find the indicated quantity. -a5, when a10 = 37 and a15 = 57

Find fraction notation. - $0.0 \overline { 1 }$

Solve the problem. -During a five-year period, a company doubles its profits each year. If the profits at the end of the fifth year are $224,000, then what are the profits for each of the first four years?

Find the indicated term of the sequence. - $a _ { n } = ( 5 n - 7 ) ( 6 n - 5 ) ; a _ { 12 }$

Find the common ratio. -64, 32, 16, 8, 4, . . .

Solve. -A ball is dropped from a height of $4 \mathrm {~m}$ . On each upward bounce the ball returns to $\frac { 3 } { 4 }$ of its previous height. Find the total distance the ball travels in an upward direction before coming to rest.

Solve. -In how many distinguishable ways can the letters of the word WINDOW be arranged?

Find the indicated term of the binomial expansion. -7 th term; $( x + 2 y ) ^ { 10 }$

Expand. - $( x - \sqrt { 7 } ) ^ { 5 }$

Find the nth, or general, term. -1, 7, 49, . . .

Expand. - $\left( - 3 x ^ { 2 } - y \right) ^ { 5 }$

Find the sum. - $\sum _ { k = 1 } ^ { 4 } ( 3 k - 2 )$

Expand. - $\left( \frac { 1 } { 3 } x + 2 \right) ^ { 3 }$

Find the indicated term of the sequence. - $a _ { n } = 5 n ^ { 2 } ( 15 n - 118 ) ; a _ { 6 }$

Use mathematical induction to prove the following. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Find the first term and the common difference. - $\frac { 7 } { 12 } , \frac { 1 } { 3 } , \frac { 1 } { 12 } , - \frac { 1 } { 6 } , \ldots$

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Exam 8: Sequences, Series, and Combinatorics

Use mathematical induction to prove the following. - 2n>n2 ^ { n } > n2n>n

Find the first term and the common difference. - 6,9,12,15,…6,9,12,15 , \ldots6,9,12,15,…