Question 1

Expand the binomial by using the binomial theorem.
-$$( 5 + y ) ^ { 4 }$$

Accepted Answer

A)  $625 + y ^ { 4 }$ 
B)  $625 + 4 y + 6 y ^ { 2 } + 4 y ^ { 3 } + y ^ { 4 }$ 
C)  $625 + 125 y + 25 y ^ { 2 } + 5 y ^ { 3 } + y ^ { 4 }$ 
D)  $625 + 500 y + 150 y ^ { 2 } + 20 y ^ { 3 } + y ^ { 4 }$ 
A)  $625 + y ^ { 4 }$ 
B)  $625 + 4 y + 6 y ^ { 2 } + 4 y ^ { 3 } + y ^ { 4 }$ 
C)  $625 + 125 y + 25 y ^ { 2 } + 5 y ^ { 3 } + y ^ { 4 }$ 
D)  $625 + 500 y + 150 y ^ { 2 } + 20 y ^ { 3 } + y ^ { 4 }$

Question 2

Use the Binomial Theorem to find the value of the number raised to the given power.
-$( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$.

Accepted Answer

A)  $1.331$ 
B)  $1.21$ 
C)  $1.001$ 
D)  $3.3$ 
A)  $1.331$ 
B)  $1.21$ 
C)  $1.001$ 
D)  $3.3$

Question 3

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-If P dollars is invested at the end of each compounding period n times per year at interest rate r, then
the value A of the annuity (in $) after t years is given by the formula
A =________ .

Accepted Answer

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

Question 4

Write a formula for the nth term of the geometric sequence.
-$1,000,200,40,8 , \ldots$

Accepted Answer

A)  $a _ { n } = 1,000 \left( \frac { 1 } { 5 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 1,000 ( - 800 ) ^ { n - 1 }$ 
C)  $a _ { n } = 1,000 ( - 800 ) ^ { n }$ 
D)  $a _ { n } = 1,000 \left( \frac { 1 } { 5 } \right) ^ { n }$ 
A)  $a _ { n } = 1,000 \left( \frac { 1 } { 5 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = 1,000 ( - 800 ) ^ { n - 1 }$ 
C)  $a _ { n } = 1,000 ( - 800 ) ^ { n }$ 
D)  $a _ { n } = 1,000 \left( \frac { 1 } { 5 } \right) ^ { n }$

Question 5

The nth term of a sequence is given. Find the indicated term.
-$$c _ { n } = \frac { ( 2 n ) ! } { 5 n } ; c _ { 5 }$$

Accepted Answer

A)  8064 
B)  $3,628,800$ 
C)  145,152 
D)  $\frac { 2 } { 5 }$ 
A)  8064 
B)  $3,628,800$ 
C)  145,152 
D)  $\frac { 2 } { 5 }$

Question 6

Write the sum using summation notation.
-$$- 3 + 9 - 27 + 81$$

Accepted Answer

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

Question 7

Use mathematical induction to prove the given statement for all positive integers n and real numbers x
and y.
-$$\left( \frac { x } { y } \right) ^ { n } = \frac { x ^ { n } } { y ^ { n } } \text { provided that } \mathrm { y } \neq 0$$

Accepted Answer

The answer of Use mathematical induction to prove the given...

Question 8

Solve the problem.
-Elizabeth must choose between two job offers. The first job offers $42,000 for the first year and a $3,300 raise each year thereafter. The second job offers $48,000 for the first year and a $2,100 raise
Each year thereafter.
a. If she anticipates working for the company for 4 years, find the total amount she would earn from each Job)
b.If she anticipates working for the company for 12 years, find the total amount she would earn from Each job.

Accepted Answer

A)  a. Job 1: $204,600; Job 2: $187,800
b. Job 1: $714,600; Job 2: $721,800 
B)  a. Job 1: $187,800; Job 2: $204,600
b. Job 1: $721,800; Job 2: $714,600 
C)  a. Job 1: $189,000; Job 2: $225,000
b. Job 1: $667,800; Job 2: $833,400 
D)  a. Job 1: $225,000; Job 2: $189,000
b. Job 1: $833,400; Job 2: $667,800 
A)  a. Job 1: $204,600; Job 2: $187,800
b. Job 1: $714,600; Job 2: $721,800 
B)  a. Job 1: $187,800; Job 2: $204,600
b. Job 1: $721,800; Job 2: $714,600 
C)  a. Job 1: $189,000; Job 2: $225,000
b. Job 1: $667,800; Job 2: $833,400 
D)  a. Job 1: $225,000; Job 2: $189,000
b. Job 1: $833,400; Job 2: $667,800

Question 9

Evaluate the expression.
-$$\frac { ( n + 2 ) ! } { ( n + 4 ) ! }$$

Accepted Answer

A)  $\frac { 1 } { ( n + 4 ) ( n + 3 ) }$ 
B)  $\frac { 1 } { ( n + 4 ) }$ 
C)  $( n + 4 ) ( n + 3 )$ 
D)  $\frac { 1 } { ( n + 3 ) ! }$ 
A)  $\frac { 1 } { ( n + 4 ) ( n + 3 ) }$ 
B)  $\frac { 1 } { ( n + 4 ) }$ 
C)  $( n + 4 ) ( n + 3 )$ 
D)  $\frac { 1 } { ( n + 3 ) ! }$

Question 10

An ________ sequence is a sequence in which consecutive terms alternate in sign.

Accepted Answer

The answer of An ________ sequence is a sequence in...

Question 11

Write the first four terms of the geometric sequence.
-$$a _ { 1 } = 40 , a _ { n } = \frac { 1 } { 5 } a _ { n - 1 }$$

Accepted Answer

A)  $8 , \frac { 8 } { 5 } , \frac { 8 } { 25 } , \frac { 8 } { 125 }$ 
B)  $40,8 , \frac { 8 } { 5 } , \frac { 8 } { 25 }$ 
C)  $40,200,1,000,5,000$ 
D)  $40 , \frac { 201 } { 5 } , \frac { 202 } { 5 } , \frac { 203 } { 5 }$ 
A)  $8 , \frac { 8 } { 5 } , \frac { 8 } { 25 } , \frac { 8 } { 125 }$ 
B)  $40,8 , \frac { 8 } { 5 } , \frac { 8 } { 25 }$ 
C)  $40,200,1,000,5,000$ 
D)  $40 , \frac { 201 } { 5 } , \frac { 202 } { 5 } , \frac { 203 } { 5 }$

Question 12

Find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each
compounding period, n times per year, at an interest rate r for t years.
-$$P = \$ 250 , n = 4 , r = 4 \% , t = 35 \mathrm { yr }$$

Accepted Answer

A)  $756.77 
B)  $75,677.48 
C)  $100,677.48 
D)  $10,415.07 
A)  $756.77 
B)  $75,677.48 
C)  $100,677.48 
D)  $10,415.07

Question 13

Solve the problem.
-Suppose that P dollars is invested at the end of each compounding period n times per year at interest rate r. Then the value A (in $) of the annuity after t years is given by $$A = \frac { P \left[ \left( 1 + \frac { r } { n } \right) ^ { n t } - 1 \right] } { \frac { r } { n } } .$$ An employee invests $150 per month in an ordinary annuity. If the interest rate is 4%, find the value
Of the annuity after 25 yr.

Accepted Answer

A)  $97,695.55 
B)  $76,121.46 
C)  $97,858.15 
D)  $77,119.43 
A)  $97,695.55 
B)  $76,121.46 
C)  $97,858.15 
D)  $77,119.43

Question 14

Solve the problem.
-The arithmetic mean (average) of two numbers c and d is given by $\bar { x } = \frac { c + d } { 2 }$ . The value $\bar { x }$ is equidistant between c and d, so the sequence $c , x , d$ is an arithmetic sequence. Inserting k equally
Spaced values between c and d, yields the arithmetic sequence c, $\bar { x } _ { 1 } , \bar { x } _ { 2 } , \bar { x } _ { 3 } , \bar { x } _ { 4 } , \ldots , \bar { x } _ { n }$ , d. Use this
Information to insert five means between 22 and 70

Accepted Answer

A)  The five arithmetic means between 22 and 70 are 30, 38, 46, 54, and 62. 
B)  The arithmetic mean of 22 and 70 is 46. 
C)  The common difference is 24. 
D)  The common difference is 8. 
A)  The five arithmetic means between 22 and 70 are 30, 38, 46, 54, and 62. 
B)  The arithmetic mean of 22 and 70 is 46. 
C)  The common difference is 24. 
D)  The common difference is 8.

Question 15

Choose the one alternative that best completes the statement or answers the question.
Determine whether the sequence is geometric. If so, find the value of r.
-2, -6, 18, -54, . . .

Accepted Answer

A)  geometric; r = -8 
B)  geometric; r = 2 
C)  not geometric 
D)  geometric; r = -3 
A)  geometric; r = -8 
B)  geometric; r = 2 
C)  not geometric 
D)  geometric; r = -3

Question 16

Given an infinite sequence $\left\{ a _ { n } \right\} = a _ { 1 } , a _ { 2 } , a _ { 3 }$ , … the sum of the terms of the sequence is called an
infinite________ . The notation $S _ { n }$ is called the nth ________ ________of the sequence and is called a finite series.

Accepted Answer

The answer of Given an infinite sequence \(\left\{ a _...

Question 17

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

Accepted Answer

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

Question 18

Find the number of terms of the finite arithmetic sequence.
-$12,21,30,39 , \ldots , 525$

Accepted Answer

A)  58 
B)  59 
C)  60 
D)  57 
A)  58 
B)  59 
C)  60 
D)  57

Question 19

The nth term of a sequence is given. Find the indicated term.
-$$a _ { n } = 2 ^ { n } + 9 ; a _ { 6 }$$

Accepted Answer

A)  73 
B)  576 
C)  108 
D)  21 
A)  73 
B)  576 
C)  108 
D)  21

Question 20

Solve the problem.
-Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a card numbered between 2 and 10, inclusive or a heart.

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 11 } { 13 }$ 
C)  $\frac { 10 } { 13 }$ 
D)  $\frac { 17 } { 52 }$ 
A)  $\frac { 1 } { 4 }$ 
B)  $\frac { 11 } { 13 }$ 
C)  $\frac { 10 } { 13 }$ 
D)  $\frac { 17 } { 52 }$

Expand the binomial by using the binomial theorem. - $( 5 + y ) ^ { 4 }$

Use the Binomial Theorem to find the value of the number raised to the given power. - $( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$ .

Write a formula for the nth term of the geometric sequence. - $1,000,200,40,8 , \ldots$

The nth term of a sequence is given. Find the indicated term. - $c _ { n } = \frac { ( 2 n ) ! } { 5 n } ; c _ { 5 }$

Write the sum using summation notation. - $- 3 + 9 - 27 + 81$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $\left( \frac { x } { y } \right) ^ { n } = \frac { x ^ { n } } { y ^ { n } } \text { provided that } \mathrm { y } \neq 0$

Evaluate the expression. - $\frac { ( n + 2 ) ! } { ( n + 4 ) ! }$

An ________ sequence is a sequence in which consecutive terms alternate in sign.

Write the first four terms of the geometric sequence. - $a _ { 1 } = 40 , a _ { n } = \frac { 1 } { 5 } a _ { n - 1 }$

Find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. - $P = \$ 250 , n = 4 , r = 4 \% , t = 35 \mathrm { yr }$

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is geometric. If so, find the value of r. -2, -6, 18, -54, . . .

Given an infinite sequence $\left\{ a _ { n } \right\} = a _ { 1 } , a _ { 2 } , a _ { 3 }$ , … the sum of the terms of the sequence is called an infinite________ . The notation $S _ { n }$ is called the nth of the sequence and is called a finite series.

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

Find the number of terms of the finite arithmetic sequence. - $12,21,30,39 , \ldots , 525$

The nth term of a sequence is given. Find the indicated term. - $a _ { n } = 2 ^ { n } + 9 ; a _ { 6 }$

Solve the problem. -Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a card numbered between 2 and 10, inclusive or a heart.

Equations and Inequalities

Functions and Relations

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants and Applications

Analytic Geometry

Review of Prerequisites

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Exam 8: Sequences, Series, Induction, and Probability

Expand the binomial by using the binomial theorem. - (5+y)4( 5 + y ) ^ { 4 }(5+y)4

Use the Binomial Theorem to find the value of the number raised to the given power. - (1.1)3( 1.1 ) ^ { 3 }(1.1)3 Hint: Write the expression as (1+0.1)3( 1 + 0.1 ) ^ { 3 }(1+0.1)3 .

Write a formula for the nth term of the geometric sequence. - 1,000,200,40,8,…1,000,200,40,8 , \ldots1,000,200,40,8,…

The nth term of a sequence is given. Find the indicated term. - cn=(2n)!5n;c5c _ { n } = \frac { ( 2 n ) ! } { 5 n } ; c _ { 5 }cn​=5n(2n)!​;c5​

Write the sum using summation notation. - −3+9−27+81- 3 + 9 - 27 + 81−3+9−27+81

Evaluate the expression. - (n+2)!(n+4)!\frac { ( n + 2 ) ! } { ( n + 4 ) ! }(n+4)!(n+2)!​

An ________ sequence is a sequence in which consecutive terms alternate in sign.

Write the first four terms of the geometric sequence. - a1=40,an=15an−1a _ { 1 } = 40 , a _ { n } = \frac { 1 } { 5 } a _ { n - 1 }a1​=40,an​=51​an−1​

Find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. - P=$250,n=4,r=4%,t=35yrP = \$ 250 , n = 4 , r = 4 \% , t = 35 \mathrm { yr }P=$250,n=4,r=4%,t=35yr

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is geometric. If so, find the value of r. -2, -6, 18, -54, . . .

Find the sum of the geometric series, if possible. - ∑n=1∞(−12)n\sum _ { n = 1 } ^ { \infty } \left( - \frac { 1 } { 2 } \right) ^ { n }∑n=1∞​(−21​)n

Find the number of terms of the finite arithmetic sequence. - 12,21,30,39,…,52512,21,30,39 , \ldots , 52512,21,30,39,…,525

The nth term of a sequence is given. Find the indicated term. - an=2n+9;a6a _ { n } = 2 ^ { n } + 9 ; a _ { 6 }an​=2n+9;a6​

Solve the problem. -Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a card numbered between 2 and 10, inclusive or a heart.

Equations and Inequalities

Functions and Relations

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants and Applications

Analytic Geometry

Review of Prerequisites

Filters

Expand the binomial by using the binomial theorem. - $( 5 + y ) ^ { 4 }$

Use the Binomial Theorem to find the value of the number raised to the given power. - $( 1.1 ) ^ { 3 }$ Hint: Write the expression as $( 1 + 0.1 ) ^ { 3 }$ .

Write a formula for the nth term of the geometric sequence. - $1,000,200,40,8 , \ldots$

The nth term of a sequence is given. Find the indicated term. - $c _ { n } = \frac { ( 2 n ) ! } { 5 n } ; c _ { 5 }$

Write the sum using summation notation. - $- 3 + 9 - 27 + 81$

Evaluate the expression. - $\frac { ( n + 2 ) ! } { ( n + 4 ) ! }$

Write the first four terms of the geometric sequence. - $a _ { 1 } = 40 , a _ { n } = \frac { 1 } { 5 } a _ { n - 1 }$

Find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. - $P = \$ 250 , n = 4 , r = 4 \% , t = 35 \mathrm { yr }$

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

Find the number of terms of the finite arithmetic sequence. - $12,21,30,39 , \ldots , 525$

The nth term of a sequence is given. Find the indicated term. - $a _ { n } = 2 ^ { n } + 9 ; a _ { 6 }$