Question 1

Find the sum.
-$$\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } 5 \cdot 7 ^ { \mathrm { k } - 1 }$$

Accepted Answer

A)  $- \frac { 5 } { 6 } \left( 1 - 7 ^ { n - 1 } \right)$ 
B)  $- 5 \left( 1 - 7 ^ { n } \right)$ 
C)  $- \frac { 5 } { 6 } \left( 1 - 7 ^ { n } \right)$ 
D)  $- 30 \left( 1 - 7 ^ { n } \right)$ 
A)  $- \frac { 5 } { 6 } \left( 1 - 7 ^ { n - 1 } \right)$ 
B)  $- 5 \left( 1 - 7 ^ { n } \right)$ 
C)  $- \frac { 5 } { 6 } \left( 1 - 7 ^ { n } \right)$ 
D)  $- 30 \left( 1 - 7 ^ { n } \right)$

Question 2

Find the nth term of the geometric sequence.
-$4,2,1 , \frac { 1 } { 2 } , \ldots$

Accepted Answer

A)  $a _ { n } = 4 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = \frac { 1 } { 2 } ( 4 ) ^ { n - 1 }$ 
C)  $a _ { n } = 4 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 4 \cdot 2 ^ { n - 1 }$ 
A)  $a _ { n } = 4 \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$ 
B)  $a _ { n } = \frac { 1 } { 2 } ( 4 ) ^ { n - 1 }$ 
C)  $a _ { n } = 4 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 4 \cdot 2 ^ { n - 1 }$

Question 3

Expand the expression using the Binomial Theorem.
-$$( x - 7 ) ^ { 5 }$$

Accepted Answer

A)  $x ^ { 5 } - 35 x ^ { 4 } + 980 x ^ { 3 } - 6,860 x ^ { 2 } + 12,005 x - 16,807$ 
B)  $x ^ { 5 } - 35 x ^ { 4 } + 490 x ^ { 3 } - 3,430 x ^ { 2 } + 12,005 x - 7$ 
C)  $x ^ { 5 } - 35 x ^ { 4 } + 490 x ^ { 3 } - 3,430 x ^ { 2 } + 12,005 x - 16,807$ 
D)  $x ^ { 5 } - 35 x ^ { 4 } + 980 x ^ { 3 } - 6,860 x ^ { 2 } + 12,005 x - 7$ 
A)  $x ^ { 5 } - 35 x ^ { 4 } + 980 x ^ { 3 } - 6,860 x ^ { 2 } + 12,005 x - 16,807$ 
B)  $x ^ { 5 } - 35 x ^ { 4 } + 490 x ^ { 3 } - 3,430 x ^ { 2 } + 12,005 x - 7$ 
C)  $x ^ { 5 } - 35 x ^ { 4 } + 490 x ^ { 3 } - 3,430 x ^ { 2 } + 12,005 x - 16,807$ 
D)  $x ^ { 5 } - 35 x ^ { 4 } + 980 x ^ { 3 } - 6,860 x ^ { 2 } + 12,005 x - 7$

Question 4

Find the indicated term of the sequence.
-The ninth term of the arithmetic sequence $- 14 \sqrt { 3 } , - 9 \sqrt { 3 } , - 4 \sqrt { 3 } , \ldots$

Accepted Answer

A)  $26 \sqrt { 3 }$ 
B)  $- 59 \sqrt { 3 }$ 
C)  $31 \sqrt { 3 }$ 
D)  $- 54 \sqrt { 3 }$ 
A)  $26 \sqrt { 3 }$ 
B)  $- 59 \sqrt { 3 }$ 
C)  $31 \sqrt { 3 }$ 
D)  $- 54 \sqrt { 3 }$

Question 5

Find the sum of the sequence.
-$$\sum _ { \mathrm { k } = 5 } ^ { 8 } \frac { 1 } { \mathrm { k } + 4 }$$

Accepted Answer

A)  $- \frac { 197 } { 210 }$ 
B)  $\frac { 624 } { 625 }$ 
C)  42 
D)  $\frac { 763 } { 1980 }$ 
A)  $- \frac { 197 } { 210 }$ 
B)  $\frac { 624 } { 625 }$ 
C)  42 
D)  $\frac { 763 } { 1980 }$

Question 6

The sequence is defined recursively. Write the first four terms.
-$$a _ { 1 } = 2 \text { and } a _ { n } = a _ { n - 1 } - 4 \text { for } n \geq 2$$

Accepted Answer

A)  -4, -8, -12, -16 
B)  2, 6 , 10 , 14 
C)  2, 0 , -4 , -8 
D)  2, -2, -6, -10 
A)  -4, -8, -12, -16 
B)  2, 6 , 10 , 14 
C)  2, 0 , -4 , -8 
D)  2, -2, -6, -10

Question 7

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$7 + 14 + 21 + \ldots + 7 n = \frac { 7 n ( n + 1 ) } { 2 }$$

Accepted Answer

First, we show the statement is true whe

Question 8

Solve the problem.
-Initially, a pendulum swings through an arc of 3 feet. On each successive swing, the length of the arc is 0.8 of the
previous length. After 10 swings, what total length will the pendulum have swung (to the nearest tenth of a
foot)?

Accepted Answer

approximat

Question 9

Find the indicated term of the sequence.
-The fourteenth term of the arithmetic sequence 25, 19, 13, ...

Accepted Answer

A)  -59 
B)  103 
C)  -53 
D)  -78 
A)  -59 
B)  103 
C)  -53 
D)  -78

Question 10

Express the sum using summation notation.
-$$\frac { 1 } { 4 } + \frac { 2 } { 5 } + \frac { 1 } { 2 } + \cdots + \frac { 4 } { 5 }$$

Accepted Answer

A) 
$\sum _ { k = 1 } ^ { n } \frac { k } { k + 3 }$ 
B) 
$\sum _ { k = 1 } ^ { 12 } \frac { k } { k + 3 }$ 
C) 
$\sum _ { k = 0 } ^ { 12 } \frac { k } { k + 3 }$ 
D) 
$\sum _ { k = 3 } ^ { 12 } \frac { k } { k + 1 }$ 
A) 
$\sum _ { k = 1 } ^ { n } \frac { k } { k + 3 }$ 
B) 
$\sum _ { k = 1 } ^ { 12 } \frac { k } { k + 3 }$ 
C) 
$\sum _ { k = 0 } ^ { 12 } \frac { k } { k + 3 }$ 
D) 
$\sum _ { k = 3 } ^ { 12 } \frac { k } { k + 1 }$

Question 11

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are
given.
-$$\begin{array} { l l } 
\mathrm { a } = 71 ; & \mathrm { d } = - 4 \
\mathrm { a } _ { \mathrm { n } } = ? ; & \mathrm { a } _ { 6 } = ?
\end{array}$$

Accepted Answer

A)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 23$ 
B)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 51$ 
C)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 51$ 
D)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 23$ 
A)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 23$ 
B)  $a _ { n } = 75 - 4 n ; a _ { 6 } = 51$ 
C)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 51$ 
D)  $a _ { n } = 71 - 4 n ; a _ { 6 } = 23$

Question 12

Solve the problem.
-Joytown has a present population of 40,000 and the population is increasing by 2.5% each year. How long will it take for the population to double? Round your answer to the nearest year.

Accepted Answer

A)  29 years 
B)  41 years 
C)  40 years 
D)  28 years 
A)  29 years 
B)  41 years 
C)  40 years 
D)  28 years

Question 13

The sequence is defined recursively. Write the first four terms.
-$$a _ { 1 } = 135 \text { and } a _ { n + 1 } = \frac { 1 } { 3 } \left( a _ { n } \right) \text { for } n \geq 2$$

Accepted Answer

The answer of The sequence is defined recursively. Write the...

Question 14

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

Accepted Answer

First, we show that the statement is tru

Question 15

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-Show that the formula $$2 + 4 + 6 + 8 + \cdots + 2 n = n ^ { 2 } + n + 3$$ obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some natural
number k, it is also true for the next natural number k + 1. Then show that the formula is false for n = 1.

Accepted Answer

Assume the statement is true for some na

Question 16

Find the sum of the arithmetic sequence.
-2 + 4 + 6 + ... + 1,836

Accepted Answer

A)  841,806 
B)  842,724 
C)  844,561 
D)  843,642 
A)  841,806 
B)  842,724 
C)  844,561 
D)  843,642

Question 17

Determine whether the sequence is geometric.
-3, 1, -1, -3, -5, ...

Accepted Answer

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

Question 18

Find the sum of the arithmetic sequence.
-{6n + 8}, n = 46

Accepted Answer

A)  6,969 
B)  7,130 
C)  6,854 
D)  6,716 
A)  6,969 
B)  7,130 
C)  6,854 
D)  6,716

Question 19

Solve.
-A town has a population of 20,000 people and is increasing by 10% every year. What will the population be at the end of 4 years?

Accepted Answer

A)  26,620 people 
B)  28,000 people 
C)  9,282 people 
D)  29,282 people 
A)  26,620 people 
B)  28,000 people 
C)  9,282 people 
D)  29,282 people

Question 20

Find the indicated term of the sequence.
-The thirteenth term of the arithmetic sequence 0, 9, 18, ...

Accepted Answer

A)  117 
B)  96 
C)  108 
D)  126 
A)  117 
B)  96 
C)  108 
D)  126

Find the sum. - $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } 5 \cdot 7 ^ { \mathrm { k } - 1 }$

Find the nth term of the geometric sequence. - $4,2,1 , \frac { 1 } { 2 } , \ldots$

Expand the expression using the Binomial Theorem. - $( x - 7 ) ^ { 5 }$

Find the indicated term of the sequence. -The ninth term of the arithmetic sequence $- 14 \sqrt { 3 } , - 9 \sqrt { 3 } , - 4 \sqrt { 3 } , \ldots$

Find the sum of the sequence. - $\sum _ { \mathrm { k } = 5 } ^ { 8 } \frac { 1 } { \mathrm { k } + 4 }$

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = 2 \text { and } a _ { n } = a _ { n - 1 } - 4 \text { for } n \geq 2$

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $7 + 14 + 21 + \ldots + 7 n = \frac { 7 n ( n + 1 ) } { 2 }$

Solve the problem. -Initially, a pendulum swings through an arc of 3 feet. On each successive swing, the length of the arc is 0.8 of the previous length. After 10 swings, what total length will the pendulum have swung (to the nearest tenth of a foot)?

Find the indicated term of the sequence. -The fourteenth term of the arithmetic sequence 25, 19, 13, ...

Express the sum using summation notation. - $\frac { 1 } { 4 } + \frac { 2 } { 5 } + \frac { 1 } { 2 } + \cdots + \frac { 4 } { 5 }$

Find the nth term and the indicated term of the arithmetic sequence whose initial term, a, and common difference, d, are given. - =71; =-4 =?; =?

Solve the problem. -Joytown has a present population of 40,000 and the population is increasing by 2.5% each year. How long will it take for the population to double? Round your answer to the nearest year.

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = 135 \text { and } a _ { n + 1 } = \frac { 1 } { 3 } \left( a _ { n } \right) \text { for } n \geq 2$

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

Find the sum of the arithmetic sequence. -2 + 4 + 6 + ... + 1,836

Determine whether the sequence is geometric. -3, 1, -1, -3, -5, ...

Find the sum of the arithmetic sequence. -{6n + 8}, n = 46

Solve. -A town has a population of 20,000 people and is increasing by 10% every year. What will the population be at the end of 4 years?

Find the indicated term of the sequence. -The thirteenth term of the arithmetic sequence 0, 9, 18, ...

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

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Exam 11: Sequences; Induction; the Binomial Theorem

Find the sum. - ∑k=1n5⋅7k−1\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } 5 \cdot 7 ^ { \mathrm { k } - 1 }∑k=1n​5⋅7k−1

Find the nth term of the geometric sequence. - 4,2,1,12,…4,2,1 , \frac { 1 } { 2 } , \ldots4,2,1,21​,…

Expand the expression using the Binomial Theorem. - (x−7)5( x - 7 ) ^ { 5 }(x−7)5

Find the indicated term of the sequence. -The ninth term of the arithmetic sequence −143,−93,−43,…- 14 \sqrt { 3 } , - 9 \sqrt { 3 } , - 4 \sqrt { 3 } , \ldots−143​,−93​,−43​,…

Find the sum of the sequence. - ∑k=581k+4\sum _ { \mathrm { k } = 5 } ^ { 8 } \frac { 1 } { \mathrm { k } + 4 }∑k=58​k+41​

The sequence is defined recursively. Write the first four terms. - a1=2 and an=an−1−4 for n≥2a _ { 1 } = 2 \text { and } a _ { n } = a _ { n - 1 } - 4 \text { for } n \geq 2a1​=2 and an​=an−1​−4 for n≥2

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - 7+14+21+…+7n=7n(n+1)27 + 14 + 21 + \ldots + 7 n = \frac { 7 n ( n + 1 ) } { 2 }7+14+21+…+7n=27n(n+1)​