Question 1

Evaluate the sum.  $\sum _ { k = 2 } ^ { 5 } 2 k$

Accepted Answer

A) 28 
B) 5 
C) 2 
D) 56 
E) none of these 
A) 28 
B) 5 
C) 2 
D) 56 
E) none of these

Question 2

How many three-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero.
​

Accepted Answer

A) 800 
B) ​500 
C) ​900 
D) ​600 
E) ​700 
A) 800 
B) ​500 
C) ​900 
D) ​600 
E) ​700

Question 3

Select the first five terms of the sequence.(Assume that n begins with 1.) $a _ { n } = 7 + ( - 7 ) ^ { n }$

Accepted Answer

A)  $7,0,56 , - 336,2408$ 
B)  $0,56 , - 336,2408 , - 16800$ 
C)  $1,56 , - 334,2408 , - 16803$ 
D)  $0,55 , - 336,2406 , - 16800$ 
E)  $1,55 , - 334,2406 , - 16803$ 
A)  $7,0,56 , - 336,2408$ 
B)  $0,56 , - 336,2408 , - 16800$ 
C)  $1,56 , - 334,2408 , - 16803$ 
D)  $0,55 , - 336,2406 , - 16800$ 
E)  $1,55 , - 334,2406 , - 16803$

Question 4

Evaluate using Pascal's triangle.   ${ } _ { 7 } C _ { 4 }$

Accepted Answer

A)  ${ } _ { 7 } C _ { 4 } = 840$ 
B)  ${ } _ { 7 } C _ { 4 } = 70$ 
C)  ${ } _ { 7 } C _ { 4 } = 126$ 
D)  ${ } _ { 7 } C _ { 4 } = 15$ 
E)  ${ } _ { 7 } C _ { 4 } = 35$ 
A)  ${ } _ { 7 } C _ { 4 } = 840$ 
B)  ${ } _ { 7 } C _ { 4 } = 70$ 
C)  ${ } _ { 7 } C _ { 4 } = 126$ 
D)  ${ } _ { 7 } C _ { 4 } = 15$ 
E)  ${ } _ { 7 } C _ { 4 } = 35$

Question 5

A deposit of $ 5000 is made in an account that earns 4 % interest compounded quarterly.The balance in the account after n quarters is given by $A _ { n } = 5000 \left( 1 + \frac { 0.04 } { 4 } \right) ^ { n } , n = 1,2,3 , \ldots$ Find the balance in the account after 10 years by finding the 40th term of the sequence.Round to the nearest penny.

Accepted Answer

A) $ 7518.76. 
B) $ 7370.61 
C) $ 7444.32 
D) $ 7518.76 
E) $ 24,005.10 
A) $ 7518.76. 
B) $ 7370.61 
C) $ 7444.32 
D) $ 7518.76 
E) $ 24,005.10

Question 6

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 20 from 1 through 25 in increasing order. ​

Accepted Answer

A) 7 
B) 5 
C) 11 
D) 13 
E) 9 
A) 7 
B) 5 
C) 11 
D) 13 
E) 9

Question 7

Use the Binomial Theorem to expand and simplify the expression. $( 4 x + 3 y ) ^ { 4 }$

Accepted Answer

A)  $256 x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
B)  $256 x ^ { 4 } + 4 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + y ^ { 4 }$ 
C)  $256 x ^ { 4 } + 108 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
D)  $x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
E)  $256 x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 } + 81 y ^ { 4 }$ 
A)  $256 x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
B)  $256 x ^ { 4 } + 4 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + y ^ { 4 }$ 
C)  $256 x ^ { 4 } + 108 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
D)  $x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 432 x y ^ { 3 } + 81 y ^ { 4 }$ 
E)  $256 x ^ { 4 } + 768 x ^ { 3 } y + 864 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 } + 81 y ^ { 4 }$

Question 8

Which scenario(s) below should be counted using permutations?
I.the number of ways the gold, silver, and bronze medal winners can be chosen in an Olympic track-and-field event (assume that there are no ties)
II.the number of ways 3 people can be chosen to sit from among 8 people
III.the number of ways to choose 5 lottery numbers
IV.the number of ways to roll a total of 7 when rolling a pair of dice
V.the permutation may not be used in any scenarios

Accepted Answer

A) ​II only 
B) ​I only 
C) ​V only 
D) ​IV only 
E) ​III only 
A) ​II only 
B) ​I only 
C) ​V only 
D) ​IV only 
E) ​III only

Question 9

Determine whether the sequence is geometric.If so, find the common ratio.   $4,20,100,500 , \ldots$

Accepted Answer

A) Geometric sequence, $r = 4$ 
B) Geometric sequence, $r = 6$ 
C) Geometric sequence, $r = 5$ 
D) Not a geometric sequence 
E) Geometric sequence, $r = 7$ 
A) Geometric sequence, $r = 4$ 
B) Geometric sequence, $r = 6$ 
C) Geometric sequence, $r = 5$ 
D) Not a geometric sequence 
E) Geometric sequence, $r = 7$

Question 10

Prove the inequality for the indicated integer values of n. $\left( \frac { 6 } { 5 } \right) ^ { n } > n , n \geq 15$

Accepted Answer

1) @#IMG-DLM& . The statement is true fo

Question 11

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } - 3 \left( - \frac { 1 } { 5 } \right) ^ { n }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 5 } { 2 }$ 
C)  $- \frac { 5 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
E) undefined 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 5 } { 2 }$ 
C)  $- \frac { 5 } { 2 }$ 
D)  $- \frac { 1 } { 2 }$ 
E) undefined

Question 12

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65$

Accepted Answer

A)  $a _ { n } = 4 n ^ { 2 } + 5$ 
B)  $a _ { n } = 8 n + 5$ 
C)  $a _ { n } = 4 n ^ { 2 } - 8 n - 5$ 
D)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$ 
E)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$. 
A)  $a _ { n } = 4 n ^ { 2 } + 5$ 
B)  $a _ { n } = 8 n + 5$ 
C)  $a _ { n } = 4 n ^ { 2 } - 8 n - 5$ 
D)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$ 
E)  $a _ { n } = 4 n ^ { 2 } - 8 n + 5$.

Question 13

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Accepted Answer

A)  $\frac { 2 } { 81 }$ 
B)  $\frac { 242 } { 81 }$ 
C)  $\frac { 80 } { 81 }$ 
D)  $\frac { 26 } { 81 }$ 
E)  $\frac { 82 } { 81 }$ 
A)  $\frac { 2 } { 81 }$ 
B)  $\frac { 242 } { 81 }$ 
C)  $\frac { 80 } { 81 }$ 
D)  $\frac { 26 } { 81 }$ 
E)  $\frac { 82 } { 81 }$

Question 14

Use mathematical induction to solve for all positive integers n.   $4 + 8 + 12 + 16 + \ldots + 2 n = ?$

Accepted Answer

A)  $( n + 3 ) n$ 
B)  $\frac { n ( n + 2 ) } { 2 }$ 
C)  $( n + 2 ) n$ 
D)  $\left( \frac { n ( n + 2 ) } { 2 } \right) ^ { 2 }$ 
E)  $\frac { n ( n + 1 ) ( n + 2 ) } { 6 }$ 
A)  $( n + 3 ) n$ 
B)  $\frac { n ( n + 2 ) } { 2 }$ 
C)  $( n + 2 ) n$ 
D)  $\left( \frac { n ( n + 2 ) } { 2 } \right) ^ { 2 }$ 
E)  $\frac { n ( n + 1 ) ( n + 2 ) } { 6 }$

Question 15

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first, second, and third (no ties)?

Accepted Answer

A) 1,728 
B) 6 
C) 36 
D) 3 
E) 1,320 
A) 1,728 
B) 6 
C) 36 
D) 3 
E) 1,320

Question 16

Find the rational number representation of the repeating decimal. $0.725$

Accepted Answer

A)  $\frac { 725 } { 99 }$ 
B)  $\frac { 999 } { 725 }$ 
C)  $\frac { 725 } { 9 }$ 
D)  $\frac { 725 } { 999 }$ 
E)  $\frac { 725 } { 9999 }$ 
A)  $\frac { 725 } { 99 }$ 
B)  $\frac { 999 } { 725 }$ 
C)  $\frac { 725 } { 9 }$ 
D)  $\frac { 725 } { 999 }$ 
E)  $\frac { 725 } { 9999 }$

Question 17

Use the Binomial Theorem to expand and simplify the expression.   $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$

Accepted Answer

A)  $u ^ { 3 } + 25 u ^ { 12 / 5 } - 1250 u ^ { 9 / 5 } - 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
B)  $u ^ { 3 } + 250 u ^ { 12 / 5 } + 25 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
C)  $u ^ { 3 } + 250 u ^ { 12 / 5 } + 1250 u ^ { 9 / 5 } + 3125 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
D)  $u ^ { 3 } + 25 u ^ { 12 / 5 } + 3125 u ^ { 9 / 5 } + 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 1250$ 
E)  $u ^ { 3 } + 25 u ^ { 12 / 5 } + 250 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
A)  $u ^ { 3 } + 25 u ^ { 12 / 5 } - 1250 u ^ { 9 / 5 } - 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
B)  $u ^ { 3 } + 250 u ^ { 12 / 5 } + 25 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
C)  $u ^ { 3 } + 250 u ^ { 12 / 5 } + 1250 u ^ { 9 / 5 } + 3125 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$ 
D)  $u ^ { 3 } + 25 u ^ { 12 / 5 } + 3125 u ^ { 9 / 5 } + 250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 1250$ 
E)  $u ^ { 3 } + 25 u ^ { 12 / 5 } + 250 u ^ { 9 / 5 } + 1250 u ^ { 6 / 5 } + 3125 u ^ { 3 / 5 } + 3125$

Question 18

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1.)  $3,5,3,5,3 , \ldots$

Accepted Answer

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

Question 19

Write the first five terms of the sequence.Determine whether the sequence is arithmetic.If so, find the common difference.(Assume that n begins with 1.) ​
An = 6 + 4n
​

Accepted Answer

A) 38, 34, 30, 26, 22Arithmetic sequenced = -4 
B) 22, 26, 30, 34, 38Arithmetic sequenced = 4 
C) 10, 14, 18, 22, 26Arithmetic sequenced = -4 
D) 26, 22, 18, 14, 10Arithmetic sequenced = -4 
E) 10, 14, 18, 22, 26Arithmetic sequenced = 4 
A) 38, 34, 30, 26, 22Arithmetic sequenced = -4 
B) 22, 26, 30, 34, 38Arithmetic sequenced = 4 
C) 10, 14, 18, 22, 26Arithmetic sequenced = -4 
D) 26, 22, 18, 14, 10Arithmetic sequenced = -4 
E) 10, 14, 18, 22, 26Arithmetic sequenced = 4

Question 20

Find the probability for the experiment of tossing a coin four times.Use the sample space  
 S = {HHHH,HHHT,HHTH,HHTT,HTHH,HTTH,HTTT,THHH,THHH,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT}
 
The probability of getting exactly one head.

Accepted Answer

A)   $\frac { 8 } { 3 }$ 
B)   $\frac { 1 } { 4 }$ 
C)   $\frac { 3 } { 8 }$ 
D)  4 
E)  $\frac { 1 } { 2 }$ 
A)   $\frac { 8 } { 3 }$ 
B)   $\frac { 1 } { 4 }$ 
C)   $\frac { 3 } { 8 }$ 
D)  4 
E)  $\frac { 1 } { 2 }$

Evaluate the sum. $\sum _ { k = 2 } ^ { 5 } 2 k$

How many three-digit numbers can be formed under the following condition? The leading digit cannot be zero.

Select the first five terms of the sequence.(Assume that n begins with 1.) $a _ { n } = 7 + ( - 7 ) ^ { n }$

Evaluate using Pascal's triangle. ${ } _ { 7 } C _ { 4 }$

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 20 from 1 through 25 in increasing order.

Use the Binomial Theorem to expand and simplify the expression. $( 4 x + 3 y ) ^ { 4 }$

Determine whether the sequence is geometric.If so, find the common ratio. $4,20,100,500 , \ldots$

Prove the inequality for the indicated integer values of n. $\left( \frac { 6 } { 5 } \right) ^ { n } > n , n \geq 15$

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } - 3 \left( - \frac { 1 } { 5 } \right) ^ { n }$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Use mathematical induction to solve for all positive integers n. $4 + 8 + 12 + 16 + \ldots + 2 n = ?$

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first, second, and third (no ties)?

Find the rational number representation of the repeating decimal. $0.725$

Use the Binomial Theorem to expand and simplify the expression. $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1.) $3,5,3,5,3 , \ldots$

Write the first five terms of the sequence.Determine whether the sequence is arithmetic.If so, find the common difference.(Assume that n begins with 1.) A_n = 6 + 4n

Find the probability for the experiment of tossing a coin four times.Use the sample space S = {HHHH,HHHT,HHTH,HHTT,HTHH,HTTH,HTTT,THHH,THHH,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT} The probability of getting exactly one head.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Exam 9: Sequences Series and Probability

Evaluate the sum. ∑k=252k\sum _ { k = 2 } ^ { 5 } 2 k∑k=25​2k

How many three-digit numbers can be formed under the following condition? ​ The leading digit cannot be zero. ​

Select the first five terms of the sequence.(Assume that n begins with 1.) an=7+(−7)na _ { n } = 7 + ( - 7 ) ^ { n }an​=7+(−7)n

Evaluate using Pascal's triangle. 7C4{ } _ { 7 } C _ { 4 }7​C4​

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 20 from 1 through 25 in increasing order. ​

Use the Binomial Theorem to expand and simplify the expression. (4x+3y)4( 4 x + 3 y ) ^ { 4 }(4x+3y)4

Determine whether the sequence is geometric.If so, find the common ratio. 4,20,100,500,…4,20,100,500 , \ldots4,20,100,500,…

Prove the inequality for the indicated integer values of n. (65)n>n,n≥15\left( \frac { 6 } { 5 } \right) ^ { n } > n , n \geq 15(56​)n>n,n≥15

Find the sum of the infinite geometric series. ∑n=0∞−3(−15)n\sum _ { n = 0 } ^ { \infty } - 3 \left( - \frac { 1 } { 5 } \right) ^ { n }∑n=0∞​−3(−51​)n

Find a quadratic model for the sequence with the indicated terms. a0=5,a2=5,a5=65a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65a0​=5,a2​=5,a5​=65

Find the indicated partial sum of the series. ∑i=1∞2(13)i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }∑i=1∞​2(31​)i fourth partial sum

Use mathematical induction to solve for all positive integers n. 4+8+12+16+…+2n=?4 + 8 + 12 + 16 + \ldots + 2 n = ?4+8+12+16+…+2n=?

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first, second, and third (no ties)?

Find the rational number representation of the repeating decimal. 0.7250.7250.725

Use the Binomial Theorem to expand and simplify the expression. (u3/5+5)5\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }(u3/5+5)5

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1.) 3,5,3,5,3,…3,5,3,5,3 , \ldots3,5,3,5,3,…

Write the first five terms of the sequence.Determine whether the sequence is arithmetic.If so, find the common difference.(Assume that n begins with 1.) ​ An = 6 + 4n ​

Find the probability for the experiment of tossing a coin four times.Use the sample space S = {HHHH,HHHT,HHTH,HHTT,HTHH,HTTH,HTTT,THHH,THHH,THHT,THTH,THTT,TTHH,TTHT,TTTH,TTTT} The probability of getting exactly one head.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Topics In Analytic Geometry

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Evaluate the sum. $\sum _ { k = 2 } ^ { 5 } 2 k$

How many three-digit numbers can be formed under the following condition? The leading digit cannot be zero.

Select the first five terms of the sequence.(Assume that n begins with 1.) $a _ { n } = 7 + ( - 7 ) ^ { n }$

Evaluate using Pascal's triangle. ${ } _ { 7 } C _ { 4 }$

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 20 from 1 through 25 in increasing order.

Use the Binomial Theorem to expand and simplify the expression. $( 4 x + 3 y ) ^ { 4 }$

Determine whether the sequence is geometric.If so, find the common ratio. $4,20,100,500 , \ldots$

Prove the inequality for the indicated integer values of n. $\left( \frac { 6 } { 5 } \right) ^ { n } > n , n \geq 15$

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } - 3 \left( - \frac { 1 } { 5 } \right) ^ { n }$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 5 , a _ { 2 } = 5 , a _ { 5 } = 65$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Use mathematical induction to solve for all positive integers n. $4 + 8 + 12 + 16 + \ldots + 2 n = ?$

Find the rational number representation of the repeating decimal. $0.725$

Use the Binomial Theorem to expand and simplify the expression. $\left( u ^ { 3 / 5 } + 5 \right) ^ { 5 }$

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1.) $3,5,3,5,3 , \ldots$

Write the first five terms of the sequence.Determine whether the sequence is arithmetic.If so, find the common difference.(Assume that n begins with 1.) A_n = 6 + 4n