Question 1

Evaluate the factorial expression.
-$\frac { 8 ! } { 7 ! }$

Accepted Answer

A)  $\frac { 8 } { 7 }$ 
B)  8 
C)  1 
D)  $8 !$ 
A)  $\frac { 8 } { 7 }$ 
B)  8 
C)  1 
D)  $8 !$

Question 2

Write out the first five terms of the sequence.
-$$\left\{ ( - 1 ) ^ { n - 1 } \left( \frac { n + 1 } { 2 n - 1 } \right) \right\}$$

Accepted Answer

A)  $- 2,1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
B)  $2 , - 1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
C)  $2,1 , \frac { 4 } { 5 } , \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
D)  $- 2,1 , - \frac { 4 } { 5 } , \frac { 5 } { 7 } , - \frac { 2 } { 3 }$ 
A)  $- 2,1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
B)  $2 , - 1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
C)  $2,1 , \frac { 4 } { 5 } , \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
D)  $- 2,1 , - \frac { 4 } { 5 } , \frac { 5 } { 7 } , - \frac { 2 } { 3 }$

Question 3

A geometric sequence is given. Find the common ratio and write out the first four terms.
-$$\left\{ 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 } \right\}$$

Accepted Answer

A)  $\mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 2 ) ^ { \mathrm { n } - 1 }$
$r = 2 ; 2,4,8,16 , \ldots$ 
B)  $a _ { n } = 2 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$
$r = \frac { 1 } { 4 } ; 2 , \frac { 1 } { 2 } , \frac { 1 } { 8 } , \frac { 1 } { 32 } , \ldots$ 
C)  $a _ { n } = 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$
$\mathrm { r } = \frac { 1 } { 2 } ; 2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots$ 
D)  $a _ { n } = \frac { 1 } { 2 } ( 2 ) ^ { n - 1 }$
$r = \frac { 1 } { 2 } ; 2,4,8,16 , \ldots$ 
A)  $\mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 2 ) ^ { \mathrm { n } - 1 }$
$r = 2 ; 2,4,8,16 , \ldots$ 
B)  $a _ { n } = 2 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$
$r = \frac { 1 } { 4 } ; 2 , \frac { 1 } { 2 } , \frac { 1 } { 8 } , \frac { 1 } { 32 } , \ldots$ 
C)  $a _ { n } = 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$
$\mathrm { r } = \frac { 1 } { 2 } ; 2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots$ 
D)  $a _ { n } = \frac { 1 } { 2 } ( 2 ) ^ { n - 1 }$
$r = \frac { 1 } { 2 } ; 2,4,8,16 , \ldots$

Question 4

Solve the problem.
-The population of a town is increasing by 300 inhabitants each year. If its population at the beginning of 1990 was 26,587, what was its population at the beginning of 1996?

Accepted Answer

A)  318,954 inhabitants 
B)  159,477 inhabitants 
C)  28,087 inhabitants 
D)  28,387 inhabitants 
A)  318,954 inhabitants 
B)  159,477 inhabitants 
C)  28,087 inhabitants 
D)  28,387 inhabitants

Question 5

Solve the problem.
-A ball is dropped from a height of 25 feet. Each time it strikes the ground, it bounces up to 0.7 of the previous
height. The total distance the ball has travelled before the second bounce is 25 + 2(25 · 0.7) feet, and the total
distance the ball has travelled before bounce n + 1 is $$25 + \sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } 50 \left( 0.7 ^ { \mathrm { k } } \right) \text { feet. }$$ Use facts about infinite geometric series to calculate the total distance the ball has travelled by the time it has stopped
bouncing.

Accepted Answer

The answer of Solve the problem.
-A ball is dropped from...

Question 6

Find the sum of the sequence.
-$$\sum _ { k = 2 } ^ { 4 } k ( k + 5 )$$

Accepted Answer

A)  74 
B)  80 
C)  33 
D)  50 
A)  74 
B)  80 
C)  33 
D)  50

Question 7

The sequence is defined recursively. Write the first four terms.
-$$a _ { 1 } = z ^ { \prime } a _ { n } = a _ { n - 1 } + U$$

Accepted Answer

A)  $U , U + z , U + 2 z , U + 3 z$ 
B)  $z , z + U , z + 2 U , z + 3 U$ 
C)  $\mathrm { z } , \mathrm { U } , 2 \mathrm { U } , 3 \mathrm { U }$ 
D)  $z , z - U , z - 2 U , z - 3 U$ 
A)  $U , U + z , U + 2 z , U + 3 z$ 
B)  $z , z + U , z + 2 U , z + 3 U$ 
C)  $\mathrm { z } , \mathrm { U } , 2 \mathrm { U } , 3 \mathrm { U }$ 
D)  $z , z - U , z - 2 U , z - 3 U$

Question 8

Find the nth term of the geometric sequence.
--8, -16, -32, -64, -128 A) $a _ { n } = - 8 \cdot 2 ^ { n }$

Accepted Answer

B)  $a _ { n } = a _ { 1 } + 2 ^ { n }$ 
C)  $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$ 
D)  $a _ { n } = - 8 \cdot 2 n$ 
B)  $a _ { n } = a _ { 1 } + 2 ^ { n }$ 
C)  $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$ 
D)  $a _ { n } = - 8 \cdot 2 n$

Question 9

The given pattern continues. Write down the nth term of the sequence suggested by the pattern.
-2 , 4 , 8 , 16 , 32 , ... A) $a _ { n } = 2 + 2 ( n - 1 )$

Accepted Answer

B)  $a _ { n } = 2 n - 1 + 1$ 
C)  $a _ { n } = 2 n$ 
D)  $a _ { n } = 2 n$ 
B)  $a _ { n } = 2 n - 1 + 1$ 
C)  $a _ { n } = 2 n$ 
D)  $a _ { n } = 2 n$

Question 10

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
-$$\mathrm { a } = 7 ; \mathrm { r } = - 2$$

Accepted Answer

A)  $a _ { 5 } = - 56 ; a _ { n } = 7 \cdot ( - 2 ) ^ { n }$ 
B)  $\mathrm { a } _ { 5 } = 112 ; \mathrm { a } _ { \mathrm { n } } = 7 \cdot ( - 2 ) ^ { \mathrm { n } }$ 
C)  $\mathrm { a } _ { 5 } = 112 ; \mathrm { a } _ { \mathrm { n } } = 7 \cdot ( - 2 ) ^ { \mathrm { n } - 1 }$ 
D)  $a _ { 5 } = - 56 ; a _ { n } = 7 \cdot ( - 2 ) ^ { n - 1 }$ 
A)  $a _ { 5 } = - 56 ; a _ { n } = 7 \cdot ( - 2 ) ^ { n }$ 
B)  $\mathrm { a } _ { 5 } = 112 ; \mathrm { a } _ { \mathrm { n } } = 7 \cdot ( - 2 ) ^ { \mathrm { n } }$ 
C)  $\mathrm { a } _ { 5 } = 112 ; \mathrm { a } _ { \mathrm { n } } = 7 \cdot ( - 2 ) ^ { \mathrm { n } - 1 }$ 
D)  $a _ { 5 } = - 56 ; a _ { n } = 7 \cdot ( - 2 ) ^ { n - 1 }$

Question 11

Find the indicated coefficient or term.
-The coefficient of x in the expansion of (7x + 3)3

Accepted Answer

A)  441 
B)  189 
C)  378 
D)  9 
A)  441 
B)  189 
C)  378 
D)  9

Question 12

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of
summation.
-7 + 11 + 15 + 19 + ... + 47 A) $\sum _ { k = 0 } ^ { 32 } 7 + 4 k$

Accepted Answer

B)  $\sum _ { k = 1 } ^ { 10 } 7 + 4 k$ 
C)  $\sum _ { k = 1 } ^ { 10 } 7 k + 4$ 
D)  $\sum _ { k = 0 } ^ { 10 } 7 + 4 k$ 
B)  $\sum _ { k = 1 } ^ { 10 } 7 + 4 k$ 
C)  $\sum _ { k = 1 } ^ { 10 } 7 k + 4$ 
D)  $\sum _ { k = 0 } ^ { 10 } 7 + 4 k$

Question 13

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$4 + 8 + 12 + \ldots + 4 n = 2 n ( n + 1 )$$

Accepted Answer

First we show that the statement is true

Question 14

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
-$4 + 12 + 36 + 108 + 324 + \cdots + 4 \cdot 3 ^ { 10 }$

Accepted Answer

A)  354,292 
B)  354,294 
C)  354,329 
D)  354,272 
A)  354,292 
B)  354,294 
C)  354,329 
D)  354,272

Question 15

Find the indicated coefficient or term.
-The 5 th term in the expansion of $( 4 x + 2 y ) ^ { 9 }$

Accepted Answer

A)  $1,032,192 x ^ { 5 } y ^ { 5 }$ 
B)  $1,032,192 x ^ { 4 } y ^ { 5 }$ 
C)  $2,064,384 x ^ { 5 } y ^ { 4 }$ 
D)  $516,096 \mathrm { x } ^ { 4 } \mathrm { y } ^ { 5 }$ 
A)  $1,032,192 x ^ { 5 } y ^ { 5 }$ 
B)  $1,032,192 x ^ { 4 } y ^ { 5 }$ 
C)  $2,064,384 x ^ { 5 } y ^ { 4 }$ 
D)  $516,096 \mathrm { x } ^ { 4 } \mathrm { y } ^ { 5 }$

Question 16

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
-6th term is -10; 15th term is -46 A) $a _ { 1 } = 10 , d = 4 , a _ { n } = a _ { n - 1 } + 4$

Accepted Answer

B)  $a _ { 1 } = - 10 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
C)  $a _ { 1 } = - 30 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
D)  $a _ { 1 } = 10 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
B)  $a _ { 1 } = - 10 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
C)  $a _ { 1 } = - 30 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
D)  $a _ { 1 } = 10 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$

Question 17

Solve the problem.
-A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats. How many seats are in the civic theater?

Accepted Answer

A)  790 seats 
B)  1070 seats 
C)  1010 seats 
D)  1092 seats 
A)  790 seats 
B)  1070 seats 
C)  1010 seats 
D)  1092 seats

Question 18

Find the sum of the arithmetic sequence.
-1 + 3 + 5 + ... + 1,109

Accepted Answer

A)  308,025 
B)  306,916 
C)  309,136 
D)  308,580 
A)  308,025 
B)  306,916 
C)  309,136 
D)  308,580

Question 19

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
-$$\mathrm { a } = 1 ; \mathrm { r } = \frac { 1 } { 2 }$$

Accepted Answer

A)  $a _ { 5 } = \frac { 1 } { 16 } ; a _ { n } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
B)  $\mathrm { a } _ { 5 } = \frac { 1 } { 32 } ; \mathrm { a } _ { \mathrm { n } } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } - 1 }$ 
C)  $a _ { 5 } = \frac { 1 } { 32 } ; a _ { n } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { n }$ 
D)  $\mathrm { a } _ { 5 } = \frac { 1 } { 16 } ; \mathrm { a } _ { \mathrm { n } } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$ 
A)  $a _ { 5 } = \frac { 1 } { 16 } ; a _ { n } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
B)  $\mathrm { a } _ { 5 } = \frac { 1 } { 32 } ; \mathrm { a } _ { \mathrm { n } } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } - 1 }$ 
C)  $a _ { 5 } = \frac { 1 } { 32 } ; a _ { n } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { n }$ 
D)  $\mathrm { a } _ { 5 } = \frac { 1 } { 16 } ; \mathrm { a } _ { \mathrm { n } } = 1 \cdot \left( \frac { 1 } { 2 } \right) ^ { \mathrm { n } }$

Question 20

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-4, 12, 36, 108, 324

Accepted Answer

A)  $$1 / 3$$ 
B)  12 
C)  3 
D)  Not geometric 
A)  $$1 / 3$$ 
B)  12 
C)  3 
D)  Not geometric

Evaluate the factorial expression. - $\frac { 8 ! } { 7 ! }$

Write out the first five terms of the sequence. - $\left\{ ( - 1 ) ^ { n - 1 } \left( \frac { n + 1 } { 2 n - 1 } \right) \right\}$

A geometric sequence is given. Find the common ratio and write out the first four terms. - $\left\{ 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 } \right\}$

Solve the problem. -The population of a town is increasing by 300 inhabitants each year. If its population at the beginning of 1990 was 26,587, what was its population at the beginning of 1996?

Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 4 } k ( k + 5 )$

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = z ^ { \prime } a _ { n } = a _ { n - 1 } + U$

Find the nth term of the geometric sequence. --8, -16, -32, -64, -128 A) $a _ { n } = - 8 \cdot 2 ^ { n }$ B) $a _ { n } = a _ { 1 } + 2 ^ { n }$ C) $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$ D) $a _ { n } = - 8 \cdot 2 n$

The given pattern continues. Write down the nth term of the sequence suggested by the pattern. -2 , 4 , 8 , 16 , 32 , ... A) $a _ { n } = 2 + 2 ( n - 1 )$ B) $a _ { n } = 2 n - 1 + 1$ C) $a _ { n } = 2 n$ D) $a _ { n } = 2 n$

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $\mathrm { a } = 7 ; \mathrm { r } = - 2$

Find the indicated coefficient or term. -The coefficient of x in the expansion of (7x + 3)3

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $4 + 8 + 12 + \ldots + 4 n = 2 n ( n + 1 )$

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $4 + 12 + 36 + 108 + 324 + \cdots + 4 \cdot 3 ^ { 10 }$

Find the indicated coefficient or term. -The 5 th term in the expansion of $( 4 x + 2 y ) ^ { 9 }$

Solve the problem. -A local civic theater has 22 seats in the first row and 21 rows in all. Each successive row contains 3 additional seats. How many seats are in the civic theater?

Find the sum of the arithmetic sequence. -1 + 3 + 5 + ... + 1,109

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $\mathrm { a } = 1 ; \mathrm { r } = \frac { 1 } { 2 }$

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -4, 12, 36, 108, 324

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

Filters

Exam 11: Sequences; Induction; the Binomial Theorem

Evaluate the factorial expression. - 8!7!\frac { 8 ! } { 7 ! }7!8!​

Write out the first five terms of the sequence. - {(−1)n−1(n+12n−1)}\left\{ ( - 1 ) ^ { n - 1 } \left( \frac { n + 1 } { 2 n - 1 } \right) \right\}{(−1)n−1(2n−1n+1​)}

A geometric sequence is given. Find the common ratio and write out the first four terms. - {2(12)n−1}\left\{ 2 \left( \frac { 1 } { 2 } \right) ^ { n - 1 } \right\}{2(21​)n−1}