Question 1

Evaluate the factorial expression.
-$\frac{6 !}{3 ! 3 !}$

Accepted Answer

A)  10 
B)  40 
C) 20 
D)  120 
A)  10 
B)  40 
C) 20 
D)  120

Question 2

Write out the first five terms of the sequence.
-$\left\{ \frac { 4 } { n ^ { 2 } } \right\}$

Accepted Answer

A)  $4,1 , \frac { 4 } { 9 } , \frac { 1 } { 4 } , \frac { 4 } { 25 }$ 
B)  $\frac { 1 } { 4 } , \frac { 2 } { 9 } , \frac { 3 } { 16 } , \frac { 4 } { 25 } , \frac { 5 } { 36 }$ 
C)  $1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 }$ 
D)  $1 , \frac { 1 } { 4 } , \frac { 1 } { 9 } , \frac { 1 } { 16 } , \frac { 1 } { 25 }$ 
A)  $4,1 , \frac { 4 } { 9 } , \frac { 1 } { 4 } , \frac { 4 } { 25 }$ 
B)  $\frac { 1 } { 4 } , \frac { 2 } { 9 } , \frac { 3 } { 16 } , \frac { 4 } { 25 } , \frac { 5 } { 36 }$ 
C)  $1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 }$ 
D)  $1 , \frac { 1 } { 4 } , \frac { 1 } { 9 } , \frac { 1 } { 16 } , \frac { 1 } { 25 }$

Question 3

Write the first four terms of the sequence whose general term is given.
-$$\left\{ \frac { - 2 ( n + 1 ) ! } { n ! } \right\}$$

Accepted Answer

A)  $- 1,0,1,2$ 
B)  $- 4 , - 3 , - \frac { 4 } { 3 } , - \frac { 5 } { 12 }$ 
C)  $- 4 , - 3 , - \frac { 8 } { 3 } , - \frac { 5 } { 2 }$ 
D)  $- 4 , - 6 , - 8 , - 10$ 
A)  $- 1,0,1,2$ 
B)  $- 4 , - 3 , - \frac { 4 } { 3 } , - \frac { 5 } { 12 }$ 
C)  $- 4 , - 3 , - \frac { 8 } { 3 } , - \frac { 5 } { 2 }$ 
D)  $- 4 , - 6 , - 8 , - 10$

Question 4

Find the nth term of the geometric sequence.
-$$3 , - \frac { 3 } { 5 } , \frac { 3 } { 25 } , - \frac { 3 } { 125 } , \frac { 3 } { 625 }$$

Accepted Answer

A)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 25 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n }$ 
D)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n + 1 }$ 
A)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 25 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n }$ 
D)  $a _ { n } = 3 \cdot \left( - \frac { 1 } { 5 } \right) ^ { n + 1 }$

Question 5

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence
with the given first term, a1, and common ratio, r.
-Find $a _ { 12 }$ when $a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }$

Accepted Answer

A)  $- \frac { 2000 } { 1594323 }$ 
B)  $\frac { 5989 } { 3 }$ 
C)  $\frac { 2000 } { 531441 }$ 
D)  $- \frac { 2000 } { 177147 }$ 
A)  $- \frac { 2000 } { 1594323 }$ 
B)  $\frac { 5989 } { 3 }$ 
C)  $\frac { 2000 } { 531441 }$ 
D)  $- \frac { 2000 } { 177147 }$

Question 6

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are
given.
-$$\begin{array} { l } 
\mathrm { a } = 71 ; \mathrm { d } = - 4 \
\mathrm { a } _ { \mathrm { n } } = ? ; \mathrm { a } _ { 6 } = ?
\end{array}$$

Accepted Answer

A)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 23$ 
B)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 51$ 
C)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 51$ 
D)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 23$ 
A)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 23$ 
B)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 51$ 
C)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 51$ 
D)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 23$

Question 7

Find the sum of the sequence.
-$$\sum _ { k = 1 } ^ { 12 } 9$$

Accepted Answer

A)  9 
B)  3 
C)  108 
D)  12 
A)  9 
B)  3 
C)  108 
D)  12

Question 8

Find the sum.
-4, 16, 64, 256, 1,024

Accepted Answer

A)  1,365 
B)  1,374 
C)  1,368 
D)  1,364 
A)  1,365 
B)  1,374 
C)  1,368 
D)  1,364

Question 9

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Show that the formula
$$2 + 4 + 6 + 8 + \cdots + 2 n = n ^ { 2 } + n + 3$$
obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some n number $k _ { t }$ it is also true for the next natural number $k + 1$. Then show that the formula is false for $n = 1$. atural

Accepted Answer

Assume the statement is true for some na

Question 10

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
-$$4 - 1 + \frac { 1 } { 4 } - \cdots$$

Accepted Answer

A)  Converges; - 1 
B)  Converges; 4 
C)  Converges; $\frac { 16 } { 5 }$ 
D)  Converges; 3 
A)  Converges; - 1 
B)  Converges; 4 
C)  Converges; $\frac { 16 } { 5 }$ 
D)  Converges; 3

Question 11

Determine whether the sequence is geometric.
-7, 4, 1, -2, -5, ...

Accepted Answer

A)  Geometric 
B)  Not geometric 
A)  Geometric 
B)  Not geometric

Question 12

Choose the one alternative that best completes the statement or answers the question.
-Joytown has a present population of 40,000 and the population is increasing by 2.5% each year. How long will it take for the population to double? Round your answer to the nearest year.

Accepted Answer

A)  29 years 
B)  41 years 
C)  40 years 
D)  28 years 
A)  29 years 
B)  41 years 
C)  40 years 
D)  28 years

Question 13

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

Accepted Answer

A)  $- 12$ 
B)  $- 4$ 
C)  $- 3$ 
D)  $- 10$ 
A)  $- 12$ 
B)  $- 4$ 
C)  $- 3$ 
D)  $- 10$

Question 14

Use a graphing utility to find the sum of each sequence.
-{ 3.4n + 8.75}, n=10

Accepted Answer

A)  231.75 
B)  192.4 
C)  42.75 
D)  274.5 
A)  231.75 
B)  192.4 
C)  42.75 
D)  274.5

Question 15

Choose the one alternative that best completes the statement or answers the question.
-A baseball player signs a contract with a starting salary of $840,000 per year and an annual increase of 4.5% beginning in the second year. What will the athlete's salary be, to the nearest dollar, in the sixth year?

Accepted Answer

A)  $1,048,864 
B)  $1,045,321 
C)  $1,046,793 
D)  $1,047,440 
A)  $1,048,864 
B)  $1,045,321 
C)  $1,046,793 
D)  $1,047,440

Question 16

Find the sum of the sequence.
-$$\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } ( k + 6 )$$

Accepted Answer

A)  2 
B)  34 
C)  $- 34$ 
D)  26 
A)  2 
B)  34 
C)  $- 34$ 
D)  26

Question 17

Write the word or phrase that best completes each statement or answers the question.
-$$\text { The coefficient of } \frac { 1 } { x } \text { in the expansion of } \left( 2 x + \frac { 1 } { x } \right) ^ { 3 }$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 18

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$$

Accepted Answer

First we show that the statement is true

Question 19

Find the sum of the sequence.
-$$\sum _ { k = 5 } ^ { 8 } \frac { 1 } { k + 4 }$$

Accepted Answer

A)  $- \frac { 197 } { 210 }$ 
B)  $\frac { 624 } { 625 }$ 
C)  42 
D)  $\frac { 763 } { 1980 }$ 
A)  $- \frac { 197 } { 210 }$ 
B)  $\frac { 624 } { 625 }$ 
C)  42 
D)  $\frac { 763 } { 1980 }$

Question 20

Expand the expression using the Binomial Theorem.
-$$( x + 1 ) ^ { 6 }$$

Accepted Answer

A)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 6$ 
B)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 1$ 
C)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 30 x ^ { 2 } + 6 x + 1$ 
D)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 360 x ^ { 2 } + 720 x + 720$ 
A)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 6$ 
B)  $x ^ { 6 } + 6 x ^ { 5 } + 15 x ^ { 4 } + 20 x ^ { 3 } + 15 x ^ { 2 } + 6 x + 1$ 
C)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 30 x ^ { 2 } + 6 x + 1$ 
D)  $x ^ { 6 } + 6 x ^ { 5 } + 30 x ^ { 4 } + 120 x ^ { 3 } + 360 x ^ { 2 } + 720 x + 720$

Evaluate the factorial expression. - $\frac{6 !}{3 ! 3 !}$

Write out the first five terms of the sequence. - $\left\{ \frac { 4 } { n ^ { 2 } } \right\}$

Write the first four terms of the sequence whose general term is given. - $\left\{ \frac { - 2 ( n + 1 ) ! } { n ! } \right\}$

Find the nth term of the geometric sequence. - $3 , - \frac { 3 } { 5 } , \frac { 3 } { 25 } , - \frac { 3 } { 125 } , \frac { 3 } { 625 }$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a1, and common ratio, r. -Find $a _ { 12 }$ when $a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }$

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are given. - =71;=-4 =?;=?

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 12 } 9$

Find the sum. -4, 16, 64, 256, 1,024

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $4 - 1 + \frac { 1 } { 4 } - \cdots$

Determine whether the sequence is geometric. -7, 4, 1, -2, -5, ...

Choose the one alternative that best completes the statement or answers the question. -Joytown has a present population of 40,000 and the population is increasing by 2.5% each year. How long will it take for the population to double? Round your answer to the nearest year.

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

Use a graphing utility to find the sum of each sequence. -{ 3.4n + 8.75}, n=10

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } ( k + 6 )$

Write the word or phrase that best completes each statement or answers the question. - $\text { The coefficient of } \frac { 1 } { x } \text { in the expansion of } \left( 2 x + \frac { 1 } { x } \right) ^ { 3 }$

Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$

Find the sum of the sequence. - $\sum _ { k = 5 } ^ { 8 } \frac { 1 } { k + 4 }$

Expand the expression using the Binomial Theorem. - $( x + 1 ) ^ { 6 }$

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 11: Sequences; Induction; the Binomial Theorem

Evaluate the factorial expression. - 6!3!3!\frac{6 !}{3 ! 3 !}3!3!6!​

Write out the first five terms of the sequence. - {4n2}\left\{ \frac { 4 } { n ^ { 2 } } \right\}{n24​}

Write the first four terms of the sequence whose general term is given. - {−2(n+1)!n!}\left\{ \frac { - 2 ( n + 1 ) ! } { n ! } \right\}{n!−2(n+1)!​}

Find the nth term of the geometric sequence. - 3,−35,325,−3125,36253 , - \frac { 3 } { 5 } , \frac { 3 } { 25 } , - \frac { 3 } { 125 } , \frac { 3 } { 625 }3,−53​,253​,−1253​,6253​

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequence with the given first term, a1, and common ratio, r. -Find a12a _ { 12 }a12​ when a1=2,000,r=−13a _ { 1 } = 2,000 , r = - \frac { 1 } { 3 }a1​=2,000,r=−31​