Question 1

Find a recursive rule for the nth term of the sequence.
-$5,11,17,23 , \ldots$

Accepted Answer

A)  $a _ { n } = a _ { n - 1 } + 7$ 
B)  $a _ { n } = a _ { n - 1 } + 6$ 
C)  $a _ { n } = a _ { n - 1 } - 6$ 
D)  $a _ { n } = a _ { n - 1 } + 12$ 
A)  $a _ { n } = a _ { n - 1 } + 7$ 
B)  $a _ { n } = a _ { n - 1 } + 6$ 
C)  $a _ { n } = a _ { n - 1 } - 6$ 
D)  $a _ { n } = a _ { n - 1 } + 12$

Question 2

Expand the binomial.
-$$( 2 x + 1 ) ^ { 4 }$$

Accepted Answer

A)  $16 x ^ { 4 } + 32 x ^ { 3 } + 24 x ^ { 2 } + 8 x + 1$ 
B)  $16 x ^ { 3 } + 32 x ^ { 2 } + 24 x + 8$ 
C)  $\left( 4 x ^ { 2 } + 2 x + 1 \right) ^ { 4 }$ 
D)  $16 x ^ { 4 } + 32 x ^ { 3 } + 24 x ^ { 2 } + 8 x + 2$ 
A)  $16 x ^ { 4 } + 32 x ^ { 3 } + 24 x ^ { 2 } + 8 x + 1$ 
B)  $16 x ^ { 3 } + 32 x ^ { 2 } + 24 x + 8$ 
C)  $\left( 4 x ^ { 2 } + 2 x + 1 \right) ^ { 4 }$ 
D)  $16 x ^ { 4 } + 32 x ^ { 3 } + 24 x ^ { 2 } + 8 x + 2$

Question 3

Find the coefficient of the given term in the binomial expansion.
-$x ^ { 4 } y ^ { 8 }$ term, $( x + y ) ^ { 12 }$

Accepted Answer

A)  4 
B)  220 
C)  1980 
D)  495 
A)  4 
B)  220 
C)  1980 
D)  495

Question 4

For the given statement $ P_{n} $, write the statements $ P_{1}, P_{k} $, and $ P_{k+1} $
-$$1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$$

Accepted Answer

\[\begin{array} { l } 
P _ { 1 } : 1 ^ {

Question 5

Solve.
-How many different three-digit numbers can be written using digits from the set {5, 6, 7, 8, 9} without any repeating digits?

Accepted Answer

A)  20 
B)  60 
C)  10 
D)  120 
A)  20 
B)  60 
C)  10 
D)  120

Question 6

Find the coefficient of the given term in the binomial expansion.
-$x ^ { 4 }$ term, $( x - 2 ) ^ { 11 }$

Accepted Answer

A)  253,440 
B)  42,240 
C)  $- 253,440$ 
D)  $- 42,240$ 
A)  253,440 
B)  42,240 
C)  $- 253,440$ 
D)  $- 42,240$

Question 7

Expand the binomial.
-$( \sqrt { x } - \sqrt { y } ) ^ { 4 }$

Accepted Answer

A)  $x ^ { 2 } - 4 x \sqrt { x y } + 6 x y - 4 y \sqrt { x y } + y ^ { 2 }$ 
B)  $- x ^ { 2 } - 4 x \sqrt { x y } - 6 x y - 4 y \sqrt { x y } - y ^ { 2 }$ 
C)  $x ^ { 2 } + 4 x \sqrt { x y } + 6 x y + 4 y \sqrt { x y } + y ^ { 2 }$ 
D)  $x ^ { 2 } + y ^ { 2 }$ 
A)  $x ^ { 2 } - 4 x \sqrt { x y } + 6 x y - 4 y \sqrt { x y } + y ^ { 2 }$ 
B)  $- x ^ { 2 } - 4 x \sqrt { x y } - 6 x y - 4 y \sqrt { x y } - y ^ { 2 }$ 
C)  $x ^ { 2 } + 4 x \sqrt { x y } + 6 x y + 4 y \sqrt { x y } + y ^ { 2 }$ 
D)  $x ^ { 2 } + y ^ { 2 }$

Question 8

Find the sum.
-$\sum _ { k = 1 } ^ { n } \left( k ^ { 3 } + k ^ { 2 } \right)$

Accepted Answer

A)  $\frac { n ( 2 n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
B)  $\frac { \mathrm { n } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } + \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 }$ 
C)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
D)  $\frac { n ^ { 3 } ( n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
A)  $\frac { n ( 2 n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
B)  $\frac { \mathrm { n } ( \mathrm { n } + 1 ) ^ { 2 } } { 4 } + \frac { \mathrm { n } ( \mathrm { n } + 1 ) ( 2 \mathrm { n } + 1 ) } { 6 }$ 
C)  $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
D)  $\frac { n ^ { 3 } ( n + 1 ) ^ { 2 } } { 4 } + \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$

Question 9

Determine whether the infinite geometric series converges. If the series converges, determine the limit.
-$$48 - 8 + \frac { 4 } { 3 } - \frac { 2 } { 9 } + \ldots$$

Accepted Answer

A)  Converges; 8880 
B)  Diverges 
C)  Converges; $\frac { 288 } { 5 }$ 
D)  Converges; $\frac { 288 } { 7 }$ 
A)  Converges; 8880 
B)  Diverges 
C)  Converges; $\frac { 288 } { 5 }$ 
D)  Converges; $\frac { 288 } { 7 }$

Question 10

Find the sum of the first n terms of the sequence.
-10, 2, -6, -14, . . . ; n = 12

Accepted Answer

A)  - 468 
B)  - 494 
C)  - 456 
D)  - 408 
A)  - 468 
B)  - 494 
C)  - 456 
D)  - 408

Question 11

Find an explicit rule for the nth term of the sequence.
-$$3 , - 12,48 , - 192 , \ldots$$

Accepted Answer

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

Question 12

Evaluate.
-$\left( \begin{array} { c } 390 \ 0 \end{array} \right)$

Accepted Answer

A)  780 
B)  1 
C)  0 
D)  390 
A)  780 
B)  1 
C)  0 
D)  390

Question 13

In how many ways can you answer the questions on an exam that consists of 8 multiple choice questions, each of which has 4 answer choices, followed by 5 true-false questions?

Accepted Answer

A)  2,097,152 
B)  2,221,982 
C)  2,084,702 
D)  2,680,257 
A)  2,097,152 
B)  2,221,982 
C)  2,084,702 
D)  2,680,257

Question 14

Find the sum of the arithmetic sequence.
-$$\frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 }$$

Accepted Answer

A)  $\frac { 29 } { 3 }$ 
B)  $\frac { 29 } { 15 }$ 
C)  $\frac { 31 } { 15 }$ 
D)  $\frac { 31 } { 3 }$ 
A)  $\frac { 29 } { 3 }$ 
B)  $\frac { 29 } { 15 }$ 
C)  $\frac { 31 } { 15 }$ 
D)  $\frac { 31 } { 3 }$

Question 15

Write the sum using summation notation, assuming the suggested pattern continues.
-$$- 10 - 6 - 2 + 2 + \ldots + 50$$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { 15 } ( - 10 + 4 n )$ 
B)  $\sum _ { n = 0 } ^ { \infty } - 40 n$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 10 + 4 n )$ 
D)  $\sum _ { n = 0 } ^ { 15 } - 40 n$ 
A)  $\sum _ { n = 0 } ^ { 15 } ( - 10 + 4 n )$ 
B)  $\sum _ { n = 0 } ^ { \infty } - 40 n$ 
C)  $\sum _ { n = 0 } ^ { \infty } ( - 10 + 4 n )$ 
D)  $\sum _ { n = 0 } ^ { 15 } - 40 n$

Question 16

Find an explicit rule for the nth term of the arithmetic sequence.
-$16,25,34,43 , \ldots$

Accepted Answer

A)  $a _ { n } = 16 + 9 ( n )$ 
B)  $a _ { n } = 16 + 9 ( n - 1 )$ 
C)  $a _ { n } = 16 - 9 ( n - 1 )$ 
D)  $a _ { n } = 16 - 9 ( n )$ 
A)  $a _ { n } = 16 + 9 ( n )$ 
B)  $a _ { n } = 16 + 9 ( n - 1 )$ 
C)  $a _ { n } = 16 - 9 ( n - 1 )$ 
D)  $a _ { n } = 16 - 9 ( n )$

Question 17

Construct a graph for the first ten terms of the sequence.
-$$a _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Find a recursive rule for the nth term of the sequence.
-$- 7,3,13,23 , \ldots$

Accepted Answer

A)  $a _ { n } = a _ { n - 1 } - 10$ 
B)  $a _ { n } = a _ { n - 1 } + 10$ 
C)  $a _ { n } = a _ { n - 1 } + 11$ 
D)  $a _ { n } = a _ { n - 1 } - 11$ 
A)  $a _ { n } = a _ { n - 1 } - 10$ 
B)  $a _ { n } = a _ { n - 1 } + 10$ 
C)  $a _ { n } = a _ { n - 1 } + 11$ 
D)  $a _ { n } = a _ { n - 1 } - 11$

Question 19

Evaluate.
-$\left( \begin{array} { l } 7 \ 3 \end{array} \right)$

Accepted Answer

A)  840 
B)  2 
C)  35 
D)  17 
A)  840 
B)  2 
C)  35 
D)  17

Question 20

Use mathematical induction to prove the statement is true for all positive integers n.
-6 + 12 + 18 + ... + 6n = 3n(n + 1)

Accepted Answer

To prove the statement 6 + 12 + 18 + ...

Find a recursive rule for the nth term of the sequence. - $5,11,17,23 , \ldots$

Expand the binomial. - $( 2 x + 1 ) ^ { 4 }$

Find the coefficient of the given term in the binomial expansion. - $x ^ { 4 } y ^ { 8 }$ term, $( x + y ) ^ { 12 }$

For the given statement $P_{n}$ , write the statements $P_{1}, P_{k}$ , and $P_{k+1}$ - $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$

Solve. -How many different three-digit numbers can be written using digits from the set {5, 6, 7, 8, 9} without any repeating digits?

Find the coefficient of the given term in the binomial expansion. - $x ^ { 4 }$ term, $( x - 2 ) ^ { 11 }$

Expand the binomial. - $( \sqrt { x } - \sqrt { y } ) ^ { 4 }$

Find the sum. - $\sum _ { k = 1 } ^ { n } \left( k ^ { 3 } + k ^ { 2 } \right)$

Determine whether the infinite geometric series converges. If the series converges, determine the limit. - $48 - 8 + \frac { 4 } { 3 } - \frac { 2 } { 9 } + \ldots$

Find the sum of the first n terms of the sequence. -10, 2, -6, -14, . . . ; n = 12

Find an explicit rule for the nth term of the sequence. - $3 , - 12,48 , - 192 , \ldots$

Evaluate. - $\left( \begin{array} { c } 390 \\ 0 \end{array} \right)$

In how many ways can you answer the questions on an exam that consists of 8 multiple choice questions, each of which has 4 answer choices, followed by 5 true-false questions?

Find the sum of the arithmetic sequence. - $\frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 }$

Write the sum using summation notation, assuming the suggested pattern continues. - $- 10 - 6 - 2 + 2 + \ldots + 50$

Find an explicit rule for the nth term of the arithmetic sequence. - $16,25,34,43 , \ldots$

Construct a graph for the first ten terms of the sequence. - $a _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }$ $Construct a graph for the first ten terms of the sequence. - a _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }$

Find a recursive rule for the nth term of the sequence. - $- 7,3,13,23 , \ldots$

Evaluate. - $\left( \begin{array} { l } 7 \\ 3 \end{array} \right)$

Use mathematical induction to prove the statement is true for all positive integers n. -6 + 12 + 18 + ... + 6n = 3n(n + 1)

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Exam 9: Discrete Mathematics

Find a recursive rule for the nth term of the sequence. - 5,11,17,23,…5,11,17,23 , \ldots5,11,17,23,…

Expand the binomial. - (2x+1)4( 2 x + 1 ) ^ { 4 }(2x+1)4

Find the coefficient of the given term in the binomial expansion. - x4y8x ^ { 4 } y ^ { 8 }x4y8 term, (x+y)12( x + y ) ^ { 12 }(x+y)12

For the given statement Pn P_{n} Pn​ , write the statements P1,Pk P_{1}, P_{k} P1​,Pk​ , and Pk+1 P_{k+1} Pk+1​ - 12+22+32+…+n2=n(n+1)(2n+1)61 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }12+22+32+…+n2=6n(n+1)(2n+1)​

Solve. -How many different three-digit numbers can be written using digits from the set {5, 6, 7, 8, 9} without any repeating digits?

Find the coefficient of the given term in the binomial expansion. - x4x ^ { 4 }x4 term, (x−2)11( x - 2 ) ^ { 11 }(x−2)11

Expand the binomial. - (x−y)4( \sqrt { x } - \sqrt { y } ) ^ { 4 }(x​−y​)4

Find the sum. - ∑k=1n(k3+k2)\sum _ { k = 1 } ^ { n } \left( k ^ { 3 } + k ^ { 2 } \right)∑k=1n​(k3+k2)

Determine whether the infinite geometric series converges. If the series converges, determine the limit. - 48−8+43−29+…48 - 8 + \frac { 4 } { 3 } - \frac { 2 } { 9 } + \ldots48−8+34​−92​+…

Find the sum of the first n terms of the sequence. -10, 2, -6, -14, . . . ; n = 12

Find an explicit rule for the nth term of the sequence. - 3,−12,48,−192,…3 , - 12,48 , - 192 , \ldots3,−12,48,−192,…

Evaluate. - (3900)\left( \begin{array} { c } 390 \\ 0 \end{array} \right)(3900​)

In how many ways can you answer the questions on an exam that consists of 8 multiple choice questions, each of which has 4 answer choices, followed by 5 true-false questions?

Find the sum of the arithmetic sequence. - 13,23,43,83,163\frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 }31​,32​,34​,38​,316​

Write the sum using summation notation, assuming the suggested pattern continues. - −10−6−2+2+…+50- 10 - 6 - 2 + 2 + \ldots + 50−10−6−2+2+…+50

Find an explicit rule for the nth term of the arithmetic sequence. - 16,25,34,43,…16,25,34,43 , \ldots16,25,34,43,…

Construct a graph for the first ten terms of the sequence. - an=(12+1n)na _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }an​=(21​+n1​)n

Find a recursive rule for the nth term of the sequence. - −7,3,13,23,…- 7,3,13,23 , \ldots−7,3,13,23,…

Evaluate. - (73)\left( \begin{array} { l } 7 \\ 3 \end{array} \right)(73​)

Use mathematical induction to prove the statement is true for all positive integers n. -6 + 12 + 18 + ... + 6n = 3n(n + 1)

Functions and Graphs

Polynomial, Power, and Rational Functions

Exponential, Logistic, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems and Matrices

Analytic Geometry in Two and Three Dimensions

Statistics and Probability

An Introduction to Calculus: Limits, Derivatives, and Integrals

Prerequisites

Filters

Find a recursive rule for the nth term of the sequence. - $5,11,17,23 , \ldots$

Expand the binomial. - $( 2 x + 1 ) ^ { 4 }$

Find the coefficient of the given term in the binomial expansion. - $x ^ { 4 } y ^ { 8 }$ term, $( x + y ) ^ { 12 }$

For the given statement $P_{n}$ , write the statements $P_{1}, P_{k}$ , and $P_{k+1}$ - $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$

Find the coefficient of the given term in the binomial expansion. - $x ^ { 4 }$ term, $( x - 2 ) ^ { 11 }$

Expand the binomial. - $( \sqrt { x } - \sqrt { y } ) ^ { 4 }$

Find the sum. - $\sum _ { k = 1 } ^ { n } \left( k ^ { 3 } + k ^ { 2 } \right)$

Determine whether the infinite geometric series converges. If the series converges, determine the limit. - $48 - 8 + \frac { 4 } { 3 } - \frac { 2 } { 9 } + \ldots$

Find an explicit rule for the nth term of the sequence. - $3 , - 12,48 , - 192 , \ldots$

Evaluate. - $\left( \begin{array} { c } 390 \\ 0 \end{array} \right)$

Find the sum of the arithmetic sequence. - $\frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 }$

Write the sum using summation notation, assuming the suggested pattern continues. - $- 10 - 6 - 2 + 2 + \ldots + 50$

Find an explicit rule for the nth term of the arithmetic sequence. - $16,25,34,43 , \ldots$

Construct a graph for the first ten terms of the sequence. - $a _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }$ $Construct a graph for the first ten terms of the sequence. - a _ { n } = \left( \frac { 1 } { 2 } + \frac { 1 } { n } \right) ^ { n }$

Find a recursive rule for the nth term of the sequence. - $- 7,3,13,23 , \ldots$

Evaluate. - $\left( \begin{array} { l } 7 \\ 3 \end{array} \right)$