Question 1

Expand the expression using the Binomial Theorem.
-$$\left( 4 x ^ { 2 } - 1 \right) ^ { 3 }$$

Accepted Answer

A)  $64 x ^ { 3 } - 48 x ^ { 2 } + 12 x - 1$ 
B)  $64 x ^ { 6 } - 48 x ^ { 4 } + 12 x ^ { 2 } - 1$ 
C)  $256 x ^ { 8 } + 64 x ^ { 6 } - 48 x ^ { 4 } + 12 x ^ { 2 } - 1$ 
D)  $\left( 16 x ^ { 4 } - 8 x ^ { 2 } + 1 \right) ^ { 3 }$ 
A)  $64 x ^ { 3 } - 48 x ^ { 2 } + 12 x - 1$ 
B)  $64 x ^ { 6 } - 48 x ^ { 4 } + 12 x ^ { 2 } - 1$ 
C)  $256 x ^ { 8 } + 64 x ^ { 6 } - 48 x ^ { 4 } + 12 x ^ { 2 } - 1$ 
D)  $\left( 16 x ^ { 4 } - 8 x ^ { 2 } + 1 \right) ^ { 3 }$

Question 2

Expand the expression using the Binomial Theorem.
-$$( 5 x + 3 ) ^ { 3 }$$

Accepted Answer

A)  $25 x ^ { 2 } + 30 x + 9$ 
B)  $25 x ^ { 6 } + 15 x ^ { 3 } + 729$ 
C)  $125 x ^ { 3 } + 225 x ^ { 2 } + 135 x + 27$ 
D)  $125 x ^ { 3 } + 225 x ^ { 2 } + 225 x + 27$ 
A)  $25 x ^ { 2 } + 30 x + 9$ 
B)  $25 x ^ { 6 } + 15 x ^ { 3 } + 729$ 
C)  $125 x ^ { 3 } + 225 x ^ { 2 } + 135 x + 27$ 
D)  $125 x ^ { 3 } + 225 x ^ { 2 } + 225 x + 27$

Question 3

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 4

Solve the problem.
-A brick staircase has a total of 19 steps The bottom step requires 141 bricks. Each successive step requires 5 less bricks than the prior one. How many bricks are required to build the staircase?

Accepted Answer

A)  3,534 bricks 
B)  1,776.5 bricks 
C)  1,824 bricks 
D)  3,648 bricks 
A)  3,534 bricks 
B)  1,776.5 bricks 
C)  1,824 bricks 
D)  3,648 bricks

Question 5

The sequence is defined recursively. Write the first four terms.
-$$a _ { 1 } = 2 , a _ { 2 } = 5 \text { and } a _ { n } = a _ { n - 2 } - 3 a _ { n - 1 } \text { for } n \geq 3$$

Accepted Answer

A)  2, 5, -1, -16 
B)  2, 5, -13, 44 
C)  2, 5, 1, 2 
D)  2, 5, 17, -46 
A)  2, 5, -1, -16 
B)  2, 5, -13, 44 
C)  2, 5, 1, 2 
D)  2, 5, 17, -46

Question 6

The given pattern continues. Write down the nth term of the sequence suggested by the pattern.
-1 , 4 , 7 , 10 , 13 , ... A) $a _ { n } = 3 n - 2$

Accepted Answer

B)  $a _ { n } = n + 3$ 
C)  $a _ { n } = 2 n - 3$ 
D)  $a _ { n } = 1 ( 3 ) ^ { n - 1 }$ 
B)  $a _ { n } = n + 3$ 
C)  $a _ { n } = 2 n - 3$ 
D)  $a _ { n } = 1 ( 3 ) ^ { n - 1 }$

Question 7

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
-$- 5 - 15 - 45 - 135 - 405 - \cdots - 5 \cdot 3 ^ { 9 }$

Accepted Answer

A)  -147,620 
B)  -147,618 
C)  -147,583 
D)  -147,640 
A)  -147,620 
B)  -147,618 
C)  -147,583 
D)  -147,640

Question 8

Find the sum.
-$$\frac { 1 } { 7 } + \frac { 3 } { 7 } + \frac { 3 ^ { 2 } } { 7 } + \frac { 3 ^ { 3 } } { 7 } + \cdots + \frac { 3 ^ { n - 1 } } { 7 }$$

Accepted Answer

A)  $- \frac { 1 } { 7 } \left( 1 - 3 ^ { n } \right)$ 
B)  $\frac { 1 } { 14 } \left( 1 - 3 ^ { n } \right)$ 
C)  $- \frac { 2 } { 7 } \left( 1 - 3 ^ { n } \right)$ 
D)  $- \frac { 1 } { 14 } \left( 1 - 3 ^ { n } \right)$ 
A)  $- \frac { 1 } { 7 } \left( 1 - 3 ^ { n } \right)$ 
B)  $\frac { 1 } { 14 } \left( 1 - 3 ^ { n } \right)$ 
C)  $- \frac { 2 } { 7 } \left( 1 - 3 ^ { n } \right)$ 
D)  $- \frac { 1 } { 14 } \left( 1 - 3 ^ { n } \right)$

Question 9

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 10

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are
given.
-$$\begin{array} { l } 
\mathrm { a } = 9 ; \mathrm { d } = - 9 \
\mathrm { a } _ { \mathrm { n } } = ? ; \mathrm { a } _ { 13 } = ?
\end{array}$$

Accepted Answer

A)  $a _ { n } = 18 - 9 n ; a _ { 13 } = - 99$ 
B)  $a _ { n } = 18 - 9 n ; a _ { 13 } = - 54$ 
C)  $a _ { n } = 9 - 9 n ; a _ { 13 } = - 99$ 
D)  $a _ { n } = 18 + 9 n ; a _ { 13 } = - 99$ 
A)  $a _ { n } = 18 - 9 n ; a _ { 13 } = - 99$ 
B)  $a _ { n } = 18 - 9 n ; a _ { 13 } = - 54$ 
C)  $a _ { n } = 9 - 9 n ; a _ { 13 } = - 99$ 
D)  $a _ { n } = 18 + 9 n ; a _ { 13 } = - 99$

Question 11

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
-$2 - 6 + 18 - 54 + 162 - \cdots + 2 \cdot ( - 3 ) ^ { 11 }$

Accepted Answer

A)  -265,726 
B)  -265,713 
C)  -265,720 
D)  -265,722 
A)  -265,726 
B)  -265,713 
C)  -265,720 
D)  -265,722

Question 12

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.
-$\text { Find a9 when } \mathrm{a}_{1}=50,000, \mathrm{r}=0.1$

Accepted Answer

A)  0.00005 
B)  0.0005 
C)  50,000.8 
D)  0.000005 
A)  0.00005 
B)  0.0005 
C)  50,000.8 
D)  0.000005

Question 13

Determine whether the sequence is geometric.
-$$- \frac { 1 } { 4 } , - \frac { 1 } { 9 } , - \frac { 1 } { 14 } , - \frac { 1 } { 19 } , \ldots$$

Accepted Answer

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

Question 14

An arithmetic sequence is given. Find the common difference and write out the first four terms.
-{9 - 6n}

Accepted Answer

A)  d = 2; 4, 6 , 8 , 10 
B)  d = -6; 3, -3, -9, -15 
C)  d = -6; 3, -1 , -7 , -13 
D)  d = -6; -6, -12, -18, -24 
A)  d = 2; 4, 6 , 8 , 10 
B)  d = -6; 3, -3, -9, -15 
C)  d = -6; 3, -1 , -7 , -13 
D)  d = -6; -6, -12, -18, -24

Question 15

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \cdots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$$

Accepted Answer

First we show that the statement is true

Question 16

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-$$\frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 } , \frac { 32 } { 3 } , \frac { 64 } { 3 }$$

Accepted Answer

A)  2 
B)  6 
C)  4 
D)  Not geometric 
A)  2 
B)  6 
C)  4 
D)  Not geometric

Question 17

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of
summation.
-13 + 16 + 19 + 22 + ... + 37 A) $\sum _ { k = 1 } ^ { 10 } 7 + 3 k$

Accepted Answer

B)  $\sum _ { k = 0 } ^ { 24 } 7 + 3 k$ 
C)  $\sum _ { k = 2 } ^ { 24 } 7 + 3 k$ 
D)  $\sum _ { k = 2 } ^ { 10 } 7 + 3 k$ 
B)  $\sum _ { k = 0 } ^ { 24 } 7 + 3 k$ 
C)  $\sum _ { k = 2 } ^ { 24 } 7 + 3 k$ 
D)  $\sum _ { k = 2 } ^ { 10 } 7 + 3 k$

Question 18

Evaluate the expression.
-$$\left( \begin{array} { c } 
10 \
5
\end{array} \right)$$

Accepted Answer

A)  252 
B)  30,240 
C)  126 
D)  504 
A)  252 
B)  30,240 
C)  126 
D)  504

Question 19

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
-$$- 30 - 5 - \frac { 5 } { 6 } - \cdots$$

Accepted Answer

A)  Converges; -30 
B)  Converges; 6 
C)  Converges; - 36 
D)  Converges; $$- \frac { 215 } { 6 }$$ 
A)  Converges; -30 
B)  Converges; 6 
C)  Converges; - 36 
D)  Converges; $$- \frac { 215 } { 6 }$$

Question 20

Express the sum using summation notation.
-$$2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \cdots + 8 ^ { 3 }$$

Accepted Answer

A) 
$\sum _ { k = 1 } ^ { 8 } k ^ { 3 }$ 
B)

$\sum _ { k = 3 } ^ { 8 } ( k - 1 ) ^ { 3 }$ 
C) 
$\sum _ { k = 2 } ^ { 8 } k ^ { 3 }$ 
D) 
$\sum _ { k = 2 } ^ { n } k ^ { 3 }$ 
A) 
$\sum _ { k = 1 } ^ { 8 } k ^ { 3 }$ 
B)

$\sum _ { k = 3 } ^ { 8 } ( k - 1 ) ^ { 3 }$ 
C) 
$\sum _ { k = 2 } ^ { 8 } k ^ { 3 }$ 
D) 
$\sum _ { k = 2 } ^ { n } k ^ { 3 }$

Expand the expression using the Binomial Theorem. - $\left( 4 x ^ { 2 } - 1 \right) ^ { 3 }$

Expand the expression using the Binomial Theorem. - $( 5 x + 3 ) ^ { 3 }$

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

Solve the problem. -A brick staircase has a total of 19 steps The bottom step requires 141 bricks. Each successive step requires 5 less bricks than the prior one. How many bricks are required to build the staircase?

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = 2 , a _ { 2 } = 5 \text { and } a _ { n } = a _ { n - 2 } - 3 a _ { n - 1 } \text { for } n \geq 3$

The given pattern continues. Write down the nth term of the sequence suggested by the pattern. -1 , 4 , 7 , 10 , 13 , ... A) $a _ { n } = 3 n - 2$ B) $a _ { n } = n + 3$ C) $a _ { n } = 2 n - 3$ D) $a _ { n } = 1 ( 3 ) ^ { n - 1 }$

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $- 5 - 15 - 45 - 135 - 405 - \cdots - 5 \cdot 3 ^ { 9 }$

Find the sum. - $\frac { 1 } { 7 } + \frac { 3 } { 7 } + \frac { 3 ^ { 2 } } { 7 } + \frac { 3 ^ { 3 } } { 7 } + \cdots + \frac { 3 ^ { n - 1 } } { 7 }$

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

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are given. - =9;=-9 =?;=?

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $2 - 6 + 18 - 54 + 162 - \cdots + 2 \cdot ( - 3 ) ^ { 11 }$

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. - $\text { Find a9 when } \mathrm{a}_{1}=50,000, \mathrm{r}=0.1$

Determine whether the sequence is geometric. - $- \frac { 1 } { 4 } , - \frac { 1 } { 9 } , - \frac { 1 } { 14 } , - \frac { 1 } { 19 } , \ldots$

An arithmetic sequence is given. Find the common difference and write out the first four terms. -{9 - 6n}

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \cdots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. - $\frac { 4 } { 3 } , \frac { 8 } { 3 } , \frac { 16 } { 3 } , \frac { 32 } { 3 } , \frac { 64 } { 3 }$

Evaluate the expression. - $\left( \begin{array} { c } 10 \\5\end{array} \right)$

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $- 30 - 5 - \frac { 5 } { 6 } - \cdots$

Express the sum using summation notation. - $2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \cdots + 8 ^ { 3 }$

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

Review

Filters

Exam 11: Sequences; Induction; the Binomial Theorem

Expand the expression using the Binomial Theorem. - (4x2−1)3\left( 4 x ^ { 2 } - 1 \right) ^ { 3 }(4x2−1)3

Expand the expression using the Binomial Theorem. - (5x+3)3( 5 x + 3 ) ^ { 3 }(5x+3)3

Find the sum of the sequence. - ∑k=1129\sum _ { k = 1 } ^ { 12 } 9∑k=112​9