Question 1

Find the common difference d for the following arithmetic sequence. 12, 7, 2, - 3, ...

Accepted Answer

A)  5 
B)  - 5 
C)  3 
D)  - 3 
E)  - 6 
A)  5 
B)  - 5 
C)  3 
D)  - 3 
E)  - 6 B

Question 2

Find the sum of the first 50 terms in an arithmetic series that has first term - 18 and 50th term 129.

Accepted Answer

A)  $2,900$ 
B)  $2,885$ 
C)  $2,655$ 
D)  $2,775$ 
E)  $2,995$ 
A)  $2,900$ 
B)  $2,885$ 
C)  $2,655$ 
D)  $2,775$ 
E)  $2,995$ D

Question 3

Find the right part of the equality. $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) =$________

Accepted Answer

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

Question 4

Find the third term in the expansion of (a - b) 27 .

Accepted Answer

A)  - 17,550a 4 b 23  
B)  17,550a 23 b 4  
C)  - 17,550a 23 b 4  
D)  17,551a 23 b 4  
E)  - 17,551a 23 b 4  
A)  - 17,550a 4 b 23  
B)  17,550a 23 b 4  
C)  - 17,550a 23 b 4  
D)  17,551a 23 b 4  
E)  - 17,551a 23 b 4

Question 5

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$ .

Accepted Answer

A)  $\frac { 20 \sqrt { 30 } } { 3 }$ 
B)  limit does not exist 
C)  $\frac { \sqrt { 6 } } { 3 } + 500$ 
D)  $\frac { \sqrt { 6 } } { 3 } - 500$ 
E)  $\frac { \sqrt { 30 } } { 75 } - 500$ 
A)  $\frac { 20 \sqrt { 30 } } { 3 }$ 
B)  limit does not exist 
C)  $\frac { \sqrt { 6 } } { 3 } + 500$ 
D)  $\frac { \sqrt { 6 } } { 3 } - 500$ 
E)  $\frac { \sqrt { 30 } } { 75 } - 500$

Question 6

For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } > n$

Accepted Answer

A)  $n \geq 25$ 
B)  $n \geq 2$ 
C)  $n \geq 23$ 
D)  $n \leq 23$ 
E)  $37 \leq n \leq 54$ 
A)  $n \geq 25$ 
B)  $n \geq 2$ 
C)  $n \geq 23$ 
D)  $n \leq 23$ 
E)  $37 \leq n \leq 54$

Question 7

Evaluate the sum. $\sum _ { n = 20 } ^ { 110 } ( 2 n - 1 )$

Accepted Answer

A)  $11,639$ 
B)  $11,689$ 
C)  $11,789$ 
D)  $11,839$ 
E)  $11,739$ 
A)  $11,639$ 
B)  $11,689$ 
C)  $11,789$ 
D)  $11,839$ 
E)  $11,739$

Question 8

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 5 } } { x - 2 }$ by direct substitution.

Accepted Answer

A)  $\frac { \sqrt { 11 } } { 4 }$ 
B)  $\frac { \sqrt { 11 } } { 2 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\sqrt { 11 }$ 
E)  $\frac { 1 } { 4 }$ 
A)  $\frac { \sqrt { 11 } } { 4 }$ 
B)  $\frac { \sqrt { 11 } } { 2 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\sqrt { 11 }$ 
E)  $\frac { 1 } { 4 }$

Question 9

Carry out the expansion. (2a + b) 9

Accepted Answer

A)  2a 9  + 2,304a 8 b + 4,608a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9  
B)  512 a 9  + 2,304a 8 b - 4,608a 7 b 2  + 5,376a 6 b 3  - 4,032a 5 b 4  + 2,016a 4 b 5  - 672a 3 b 6  + 144a 2 b 7  - 18ab 8  + b 9  
C)  512a 9  + 2,304a 8 b + 4,610a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9  
D)  512a 9  + 2,304a 8 b - 4,608a 7 b 2  + 5,376a 6 b 3  - 4,032a 5 b 4  + 2,016a 4 b 5  - 672a 3 b 6  + 144a 2 b 7  - 18ab 8  + b 9  
E)  512a 9  + 2,304a 8 b + 4,608a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9  
A)  2a 9  + 2,304a 8 b + 4,608a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9  
B)  512 a 9  + 2,304a 8 b - 4,608a 7 b 2  + 5,376a 6 b 3  - 4,032a 5 b 4  + 2,016a 4 b 5  - 672a 3 b 6  + 144a 2 b 7  - 18ab 8  + b 9  
C)  512a 9  + 2,304a 8 b + 4,610a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9  
D)  512a 9  + 2,304a 8 b - 4,608a 7 b 2  + 5,376a 6 b 3  - 4,032a 5 b 4  + 2,016a 4 b 5  - 672a 3 b 6  + 144a 2 b 7  - 18ab 8  + b 9  
E)  512a 9  + 2,304a 8 b + 4,608a 7 b 2  + 5,376a 6 b 3  + 4,032a 5 b 4  + 2,016a 4 b 5  + 672a 3 b 6  + 144a 2 b 7  + 18ab 8  + b 9

Question 10

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists).

Accepted Answer

A)  0 
B)  3 
C)  -3 
D)  limit does not exist 
E)  6 
A)  0 
B)  3 
C)  -3 
D)  limit does not exist 
E)  6

Question 11

Find the right part of the equality. $$4 + 5 + 6 + \cdots + ( n + 3 ) =$$

Accepted Answer

A)  $\frac { n ( 8 n + 7 ) } { 3 }$ 
B)  $\frac { n ( n + 7 ) } { 2 }$ 
C)  $\frac { n ( 5 n + 7 ) } { 3 }$ 
D)  $n ( 5 n - 1 )$ 
E)  $\frac { n ( n + 7 ) } { n }$ 
A)  $\frac { n ( 8 n + 7 ) } { 3 }$ 
B)  $\frac { n ( n + 7 ) } { 2 }$ 
C)  $\frac { n ( 5 n + 7 ) } { 3 }$ 
D)  $n ( 5 n - 1 )$ 
E)  $\frac { n ( n + 7 ) } { n }$

Question 12

Find the indicated term. $11,20,29,38 , \ldots , a _ { 12 }$

Accepted Answer

A)  101 
B)  110 
C)  220 
D)  55 
E)  111 
A)  101 
B)  110 
C)  220 
D)  55 
E)  111

Question 13

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$

Accepted Answer

A)  24 
B)  F1F1F1S1F1F1F10 
C)  0 
D)  12 
E)  limit does not exist 
A)  24 
B)  F1F1F1S1F1F1F10 
C)  0 
D)  12 
E)  limit does not exist

Question 14

Find the indicated term. $5,2 , - 1 , - 4 , \ldots , a _ { 20 }$

Accepted Answer

A)  $- 49$ 
B)  $- 51$ 
C)  $- 26$ 
D)  $- 52$ 
E)  $- 56$ 
A)  $- 49$ 
B)  $- 51$ 
C)  $- 26$ 
D)  $- 52$ 
E)  $- 56$

Question 15

Find the right part of the equality. $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } =$________

Accepted Answer

A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( n + 3 ) } { 4 }$ 
C)  $\frac { n ( n + 1 ) } { 2 }$ 
D)  $\frac { ( n + 1 ) ( n + 2 ) } { 3 }$ 
E)  none of the above 
A)  $\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }$ 
B)  $\frac { n ( n + 1 ) ( n + 3 ) } { 4 }$ 
C)  $\frac { n ( n + 1 ) } { 2 }$ 
D)  $\frac { ( n + 1 ) ( n + 2 ) } { 3 }$ 
E)  none of the above

Question 16

The sum of three consecutive terms in an arithmetic sequence is 30, and their product is 750. Find the three terms. (Suggestion: Let x denote the middle term and d the common difference.)

Accepted Answer

A)  5, 9, 13 
B)  4, 9, 14 
C)  5, 10, 15 
D)  1, 10, 15 
E)  6, 10, 14 
A)  5, 9, 13 
B)  4, 9, 14 
C)  5, 10, 15 
D)  1, 10, 15 
E)  6, 10, 14

Question 17

Evaluate and simplify. $\left( \begin{array} { l } 
4 \
0
\end{array} \right)$

Accepted Answer

A)  0 
B)  4 
C)  14 
D)  9 
E)  1 
A)  0 
B)  4 
C)  14 
D)  9 
E)  1

Question 18

Express as a fraction. $0 . \overline { 71 }$

Accepted Answer

A)  $\frac { 71 } { 99 }$ 
B)  $\frac { 71 } { 100 }$ 
C)  $\frac { 71 } { 110 }$ 
D)  $\frac { 71 } { 183 }$ 
E)  $\frac { 71 } { 97 }$ 
A)  $\frac { 71 } { 99 }$ 
B)  $\frac { 71 } { 100 }$ 
C)  $\frac { 71 } { 110 }$ 
D)  $\frac { 71 } { 183 }$ 
E)  $\frac { 71 } { 97 }$

Question 19

Use the position function $s ( t ) = - 16 t ^ { 2 } + 120$ to find the velocity in feet/second at time $t = 2$ seconds. The velocity at time $t = c$ seconds is given by $\lim _ { t \rightarrow c } \frac { [ s ( c ) - s ( t ) ] } { ( c - t ) }$

Accepted Answer

A)  limit does not exist 
B)  32 feet/second 
C)  64 feet/second 
D)  0 feet/second 
E)  -64 feet/second 
A)  limit does not exist 
B)  32 feet/second 
C)  64 feet/second 
D)  0 feet/second 
E)  -64 feet/second

Question 20

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 13 + y } - \sqrt { 13 } } { y }$ .

Accepted Answer

A)  $\frac { \sqrt { 13 } } { 13 }$ 
B)  limit does not exist 
C)  $\frac { \sqrt { 13 } } { 2 }$ 
D)  $\frac { \sqrt { 13 } } { 26 }$ 
E)  0 
A)  $\frac { \sqrt { 13 } } { 13 }$ 
B)  limit does not exist 
C)  $\frac { \sqrt { 13 } } { 2 }$ 
D)  $\frac { \sqrt { 13 } } { 26 }$ 
E)  0

Find the common difference d for the following arithmetic sequence. 12, 7, 2, - 3, ...

Find the sum of the first 50 terms in an arithmetic series that has first term - 18 and 50th term 129.

Find the right part of the equality. $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) =$ ________

Find the third term in the expansion of (a - b)²⁷.

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$ .

For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } > n$

Evaluate the sum. $\sum _ { n = 20 } ^ { 110 } ( 2 n - 1 )$

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 5 } } { x - 2 }$ by direct substitution.

Carry out the expansion. (2a + b)⁹

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x } (if it exists).$

Find the right part of the equality. $4 + 5 + 6 + \cdots + ( n + 3 ) =$

Find the indicated term. $11,20,29,38 , \ldots , a _ { 12 }$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$

Find the indicated term. $5,2 , - 1 , - 4 , \ldots , a _ { 20 }$

Find the right part of the equality. $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } =$ ________

The sum of three consecutive terms in an arithmetic sequence is 30, and their product is 750. Find the three terms. (Suggestion: Let x denote the middle term and d the common difference.)

Evaluate and simplify. $\left( \begin{array} { l } 4 \\0\end{array} \right)$

Express as a fraction. $0 . \overline { 71 }$

Use the position function $s ( t ) = - 16 t ^ { 2 } + 120$ to find the velocity in feet/second at time $t = 2$ seconds. The velocity at time $t = c$ seconds is given by $\lim _ { t \rightarrow c } \frac { [ s ( c ) - s ( t ) ] } { ( c - t ) }$

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 13 + y } - \sqrt { 13 } } { y }$ .

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Filters

Exam 14: Additional Topics in Algebra

Find the common difference d for the following arithmetic sequence. 12, 7, 2, - 3, ...

Find the sum of the first 50 terms in an arithmetic series that has first term - 18 and 50th term 129.

Find the right part of the equality. 1⋅2+2⋅3+3⋅4+…+n(n+1)=1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) =1⋅2+2⋅3+3⋅4+…+n(n+1)= ________

Find the third term in the expansion of (a - b) 27 .

Find lim⁡x→5[g(x)−f(x)]\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]limx→5​[g(x)−f(x)] for f(x)=4x3f ( x ) = 4 x ^ { 3 }f(x)=4x3 and g(x)=x2+53x2g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }g(x)=3x2x2+5​​ .

For what natural numbers is the following inequality true? (1.15)n>n( 1.15 ) ^ { n } > n(1.15)n>n

Evaluate the sum. ∑n=20110(2n−1)\sum _ { n = 20 } ^ { 110 } ( 2 n - 1 )∑n=20110​(2n−1)

Find lim⁡x→6x+5x−2\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 5 } } { x - 2 }limx→6​x−2x+5​​ by direct substitution.

Carry out the expansion. (2a + b) 9

Use the graph to determine lim⁡x→0x2−3xx\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }limx→0​xx2−3x​ (if it exists).

Find the right part of the equality. 4+5+6+⋯+(n+3)=4 + 5 + 6 + \cdots + ( n + 3 ) =4+5+6+⋯+(n+3)=

Find the indicated term. 11,20,29,38,…,a1211,20,29,38 , \ldots , a _ { 12 }11,20,29,38,…,a12​

Use the graph to find lim⁡x→34x2−36x−3\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }limx→3​x−34x2−36​

Find the indicated term. 5,2,−1,−4,…,a205,2 , - 1 , - 4 , \ldots , a _ { 20 }5,2,−1,−4,…,a20​

Find the right part of the equality. 12+22+32+…+n2=1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } =12+22+32+…+n2= ________

The sum of three consecutive terms in an arithmetic sequence is 30, and their product is 750. Find the three terms. (Suggestion: Let x denote the middle term and d the common difference.)

Evaluate and simplify. (40)\left( \begin{array} { l } 4 \\0\end{array} \right)(40​)

Express as a fraction. 0.71‾0 . \overline { 71 }0.71

Find lim⁡y→013+y−13y\lim _ { y \rightarrow 0 } \frac { \sqrt { 13 + y } - \sqrt { 13 } } { y }limy→0​y13+y​−13​​ .

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Roots of Polynomial Equations

Filters

Find the right part of the equality. $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) =$ ________

Find the third term in the expansion of (a - b)²⁷.

Find $\lim _ { x \rightarrow 5 } [ g ( x ) - f ( x ) ]$ for $f ( x ) = 4 x ^ { 3 }$ and $g ( x ) = \frac { \sqrt { x ^ { 2 } + 5 } } { 3 x ^ { 2 } }$ .

For what natural numbers is the following inequality true? $( 1.15 ) ^ { n } > n$

Evaluate the sum. $\sum _ { n = 20 } ^ { 110 } ( 2 n - 1 )$

Find $\lim _ { x \rightarrow 6 } \frac { \sqrt { x + 5 } } { x - 2 }$ by direct substitution.

Carry out the expansion. (2a + b)⁹

Use the graph to determine $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x }$ (if it exists). $Use the graph to determine \lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 x } { x } (if it exists).$

Find the right part of the equality. $4 + 5 + 6 + \cdots + ( n + 3 ) =$

Find the indicated term. $11,20,29,38 , \ldots , a _ { 12 }$

Use the graph to find $\lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$ $Use the graph to find \lim _ { x \rightarrow 3 } \frac { 4 x ^ { 2 } - 36 } { x - 3 }$

Find the indicated term. $5,2 , - 1 , - 4 , \ldots , a _ { 20 }$

Find the right part of the equality. $1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + \ldots + n ^ { 2 } =$ ________

Evaluate and simplify. $\left( \begin{array} { l } 4 \\0\end{array} \right)$

Express as a fraction. $0 . \overline { 71 }$

Find $\lim _ { y \rightarrow 0 } \frac { \sqrt { 13 + y } - \sqrt { 13 } } { y }$ .