Question 1

Simplify the factorial expression. $\frac { 14 ! } { 12 ! }$

Accepted Answer

A) 2184 
B) 182 
C)  $\frac { 7 } { 6 }$ 
D) 14 
E) 2730 
A) 2184 
B) 182 
C)  $\frac { 7 } { 6 }$ 
D) 14 
E) 2730

Question 2

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 7 } 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }$

Accepted Answer

A)  $\frac { 390,369 } { 31,250 }$ 
B)  $\frac { 192 } { 5 }$ 
C)  $25,999$ 
D)  $\frac { 5187 } { 3125 }$ 
E)  $\frac { 77,997 } { 15,625 }$ 
A)  $\frac { 390,369 } { 31,250 }$ 
B)  $\frac { 192 } { 5 }$ 
C)  $25,999$ 
D)  $\frac { 5187 } { 3125 }$ 
E)  $\frac { 77,997 } { 15,625 }$

Question 3

Write the rational number $0 . \overline { 81 }$ as the quotient of two integers in simplest form.

Accepted Answer

A)  $\frac { 9 } { 11 }$ 
B)  $\frac { 4 } { 11 }$ 
C)  $\frac { 5 } { 11 }$ 
D)  $\frac { 10 } { 11 }$ 
E)  $\frac { 8 } { 11 }$ 
A)  $\frac { 9 } { 11 }$ 
B)  $\frac { 4 } { 11 }$ 
C)  $\frac { 5 } { 11 }$ 
D)  $\frac { 10 } { 11 }$ 
E)  $\frac { 8 } { 11 }$

Question 4

Write the given series in summation notation. $\frac { 1 } { 5 } + \frac { 1 } { 40 } + \frac { 1 } { 135 } + \frac { 1 } { 320 } + \frac { 1 } { 625 } + \frac { 1 } { 1080 }$

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 4 } }$ 
B)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 3 } }$ 
C)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 5 } }$ 
D)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 3 } }$ 
E)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 4 } }$ 
A)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 4 } }$ 
B)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 3 } }$ 
C)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 5 i ^ { 5 } }$ 
D)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 3 } }$ 
E)  $\sum _ { i = 1 } ^ { 6 } \frac { 1 } { 3 i ^ { 4 } }$

Question 5

A heavy object (with negligible air resistance) is dropped from a plane. During the first second of fall, the object falls 17.4 meters; during the second second, it falls 52.2 meters; during the third second, it falls 87.0 meters; and during the fourth second, it falls 121.8 meters. If this pattern continues, how many meters will the object fall in 10 seconds?

Accepted Answer

A) 2505.6 meters 
B) 1409.4 meters 
C) 626.4 meters 
D) 1740.0 meters 
E) 852.6 meters 
A) 2505.6 meters 
B) 1409.4 meters 
C) 626.4 meters 
D) 1740.0 meters 
E) 852.6 meters

Question 6

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }$

Accepted Answer

A) Ratio Test is inconclusive 
B) diverges 
C) converges 
A) Ratio Test is inconclusive 
B) diverges 
C) converges

Question 7

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 3 } { 10 } \right) ^ { n }$

Accepted Answer

A) inconclusive 
B) diverges 
C) converges 
A) inconclusive 
B) diverges 
C) converges

Question 8

Write the first five terms of the power series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { 4 ^ { n } }$ .

Accepted Answer

A)  $\frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } - \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } + \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
B)  $- \frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } - \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
C)  $- \frac { ( x - 8 ) } { 4 } + \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
D)  $\frac { ( x - 8 ) } { 4 } + \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } + \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
E)  $- \frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
A)  $\frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } - \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } + \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
B)  $- \frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } - \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
C)  $- \frac { ( x - 8 ) } { 4 } + \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } - \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
D)  $\frac { ( x - 8 ) } { 4 } + \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } + \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$ 
E)  $- \frac { ( x - 8 ) } { 4 } - \frac { ( x - 8 ) ^ { 2 } } { 4 ^ { 2 } } + \frac { ( x - 8 ) ^ { 3 } } { 4 ^ { 3 } } + \frac { ( x - 8 ) ^ { 4 } } { 4 ^ { 4 } } - \frac { ( x - 8 ) ^ { 5 } } { 4 ^ { 5 } }$

Question 9

A ball is dropped from a height of 14 feet, and on each rebound it rises to $\frac { 2 } { 5 }$ its preceding height. Write an expression for the height of the nth rebound.

Accepted Answer

A)  $h _ { n } = \frac { \left( \frac { 2 } { 5 } \right) ^ { n } } { 14 }$ 
B)  $h _ { n } = 14 \left( \frac { 2 } { 5 } \right) ^ { n }$ 
C)  $h _ { n } = 14 \left( \frac { 5 } { 2 } \right) ^ { n }$ 
D)  $h _ { n } = \frac { 14 } { \left( \frac { 2 } { 5 } \right) ^ { n } }$ 
E)  $h _ { n } = \left( 14 \frac { 5 } { 2 } \right) ^ { n }$ 
A)  $h _ { n } = \frac { \left( \frac { 2 } { 5 } \right) ^ { n } } { 14 }$ 
B)  $h _ { n } = 14 \left( \frac { 2 } { 5 } \right) ^ { n }$ 
C)  $h _ { n } = 14 \left( \frac { 5 } { 2 } \right) ^ { n }$ 
D)  $h _ { n } = \frac { 14 } { \left( \frac { 2 } { 5 } \right) ^ { n } }$ 
E)  $h _ { n } = \left( 14 \frac { 5 } { 2 } \right) ^ { n }$

Question 10

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { 900 }$ . Use a symbolic algebra utility to verify your result.

Accepted Answer

A) The series converges. 
B) The series diverges. 
A) The series converges. 
B) The series diverges.

Question 11

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

Accepted Answer

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

Question 12

Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] $0 . \overline { 48 }$

Accepted Answer

A)  $\frac { 49 } { 99 }$ 
B)  $\frac { 49 } { 100 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 16 } { 33 }$ 
E)  $\frac { 12 } { 25 }$ 
A)  $\frac { 49 } { 99 }$ 
B)  $\frac { 49 } { 100 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 16 } { 33 }$ 
E)  $\frac { 12 } { 25 }$

Question 13

Determine whether the sequence is arithmetic. If so, find the common difference. 2, 1, 0, -1, -2

Accepted Answer

A) 3 
B) -1 
C) 2 
D) 1 
E) not arithmetic 
A) 3 
B) -1 
C) 2 
D) 1 
E) not arithmetic

Question 14

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 8 } { 9 } \right) ^ { n }$

Accepted Answer

A)  $81$ 
B)  $63$ 
C)  $72$ 
D) 9 
E) 8 
A)  $81$ 
B)  $63$ 
C)  $72$ 
D) 9 
E) 8

Question 15

Write the first five terms of the sequence. an = $2 - \frac { 5 } { n } - \frac { 7 } { n ^ { 2 } }$

Accepted Answer

A)  $- 10 , - \frac { 9 } { 4 } , - \frac { 4 } { 9 } , \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$ 
B)  $- 10 , - \frac { 13 } { 2 } , - \frac { 16 } { 3 } , - \frac { 19 } { 4 },$ $- \frac { 22 } { 5 }$ 
C)  $- 10 , - \frac { 19 } { 4 } , - \frac { 34 } { 9 } , - \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$ 
D)  $- 10 , \frac { 19 } { 4 } , - \frac { 34 } { 9 } , \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$ 
E)  $- 10 , \frac { 9 } { 4 } , - \frac { 4 } { 9 } , - \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$ 
A)  $- 10 , - \frac { 9 } { 4 } , - \frac { 4 } { 9 } , \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$ 
B)  $- 10 , - \frac { 13 } { 2 } , - \frac { 16 } { 3 } , - \frac { 19 } { 4 },$ $- \frac { 22 } { 5 }$ 
C)  $- 10 , - \frac { 19 } { 4 } , - \frac { 34 } { 9 } , - \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$ 
D)  $- 10 , \frac { 19 } { 4 } , - \frac { 34 } { 9 } , \frac { 55 } { 16 },$ $- \frac { 82 } { 25 }$ 
E)  $- 10 , \frac { 9 } { 4 } , - \frac { 4 } { 9 } , - \frac { 5 } { 16 },$ $\frac { 18 } { 25 }$

Question 16

Determine the convergence or divergence of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { n ^ { 2 } }$

Accepted Answer

A) converges 
B) diverges 
A) converges 
B) diverges

Question 17

Find a formula for an for the arithmetic sequence. $a _ { 3 } = 8 , a _ { 7 } = 36$

Accepted Answer

A)  $a _ { n } = - 6 + 7 n$ 
B)  $a _ { n } = 13 - 6 n$ 
C)  $a _ { n } = 7 - 6 n$ 
D)  $a _ { n } = - 6 ( 7 ) ^ { n }$ 
E)  $a _ { n } = - 13 + 7 n$ 
A)  $a _ { n } = - 6 + 7 n$ 
B)  $a _ { n } = 13 - 6 n$ 
C)  $a _ { n } = 7 - 6 n$ 
D)  $a _ { n } = - 6 ( 7 ) ^ { n }$ 
E)  $a _ { n } = - 13 + 7 n$

Question 18

Find the nth term of the geometric sequence. $- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }$

Accepted Answer

A)  $2 \left( - \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
B)  $2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
C)  $- 2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$ 
D)  $- 2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
E)  $2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$ 
A)  $2 \left( - \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
B)  $2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
C)  $- 2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$ 
D)  $- 2 \left( \frac { 5 } { 4 } \right) ^ { n - 1 }$ 
E)  $2 \left( - \frac { 4 } { 5 } \right) ^ { n - 1 }$

Question 19

Logs are stacked so that there are 17 logs in the bottom row, 16 logs in the second row from the bottom, and so on, decreasing by 1 log each row. How many logs are there in the first five rows from the bottom?

Accepted Answer

A) 117 logs 
B) 108 logs 
C) 75 logs 
D) 87 logs 
E) 125 logs 
A) 117 logs 
B) 108 logs 
C) 75 logs 
D) 87 logs 
E) 125 logs

Question 20

Write the first five terms of the geometric sequence. $a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }$

Accepted Answer

A) -1, -7, -13, -19, -25 
B) -1, 6, -36, 216, -1296 
C)  $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \frac { 1 } { 7776 }$ 
D)  $1 , - 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 }$ 
E)  $- 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 }$ 
A) -1, -7, -13, -19, -25 
B) -1, 6, -36, 216, -1296 
C)  $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \frac { 1 } { 7776 }$ 
D)  $1 , - 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 }$ 
E)  $- 1 , \frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 }$

Simplify the factorial expression. $\frac { 14 ! } { 12 ! }$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 7 } 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }$

Write the rational number $0 . \overline { 81 }$ as the quotient of two integers in simplest form.

Write the given series in summation notation. $\frac { 1 } { 5 } + \frac { 1 } { 40 } + \frac { 1 } { 135 } + \frac { 1 } { 320 } + \frac { 1 } { 625 } + \frac { 1 } { 1080 }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }$

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 3 } { 10 } \right) ^ { n }$

Write the first five terms of the power series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { 4 ^ { n } }$ .

A ball is dropped from a height of 14 feet, and on each rebound it rises to $\frac { 2 } { 5 }$ its preceding height. Write an expression for the height of the nth rebound.

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { 900 }$ . Use a symbolic algebra utility to verify your result.

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

Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] $0 . \overline { 48 }$

Determine whether the sequence is arithmetic. If so, find the common difference. 2, 1, 0, -1, -2

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 8 } { 9 } \right) ^ { n }$

Write the first five terms of the sequence. an = $2 - \frac { 5 } { n } - \frac { 7 } { n ^ { 2 } }$

Determine the convergence or divergence of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { n ^ { 2 } }$

Find a formula for an for the arithmetic sequence. $a _ { 3 } = 8 , a _ { 7 } = 36$

Find the nth term of the geometric sequence. $- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }$

Logs are stacked so that there are 17 logs in the bottom row, 16 logs in the second row from the bottom, and so on, decreasing by 1 log each row. How many logs are there in the first five rows from the bottom?

Write the first five terms of the geometric sequence. $a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }$

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Exam 16: Series and Taylor Polynomials Web

Simplify the factorial expression. 14!12!\frac { 14 ! } { 12 ! }12!14!​

Find the sum of the finite geometric sequence. ∑n=173(25)n−1\sum _ { n = 1 } ^ { 7 } 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }∑n=17​3(52​)n−1

Write the rational number 0.81‾0 . \overline { 81 }0.81 as the quotient of two integers in simplest form.

Write the given series in summation notation. 15+140+1135+1320+1625+11080\frac { 1 } { 5 } + \frac { 1 } { 40 } + \frac { 1 } { 135 } + \frac { 1 } { 320 } + \frac { 1 } { 625 } + \frac { 1 } { 1080 }51​+401​+1351​+3201​+6251​+10801​

Use the Ratio Test to determine the convergence or divergence of the series. ∑n=1∞n(85)n\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }∑n=1∞​n(58​)n

Determine the convergence or divergence of the series. ∑n=0∞(310)n\sum _ { n = 0 } ^ { \infty } \left( \frac { 3 } { 10 } \right) ^ { n }∑n=0∞​(103​)n

Write the first five terms of the power series ∑n=1∞(−1)n(x−8)n4n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { 4 ^ { n } }∑n=1∞​4n(−1)n(x−8)n​ .

A ball is dropped from a height of 14 feet, and on each rebound it rises to 25\frac { 2 } { 5 }52​ its preceding height. Write an expression for the height of the nth rebound.

Determine the convergence or divergence of the series ∑n=1∞6n900\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { 900 }∑n=1∞​9006n​ . Use a symbolic algebra utility to verify your result.

Find the sum of the infinite geometric series. ∑n=1∞(−13)n\sum _ { n = 1 } ^ { \infty } \left( - \frac { 1 } { 3 } \right) ^ { n }∑n=1∞​(−31​)n

Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] 0.48‾0 . \overline { 48 }0.48

Determine whether the sequence is arithmetic. If so, find the common difference. 2, 1, 0, -1, -2

Find the sum of the convergent series. ∑n=0∞9(89)n\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 8 } { 9 } \right) ^ { n }∑n=0∞​9(98​)n

Write the first five terms of the sequence. an = 2−5n−7n22 - \frac { 5 } { n } - \frac { 7 } { n ^ { 2 } }2−n5​−n27​

Determine the convergence or divergence of the following series. ∑n=1∞2n2\sum _ { n = 1 } ^ { \infty } \frac { 2 } { n ^ { 2 } }∑n=1∞​n22​

Find a formula for an for the arithmetic sequence. a3=8,a7=36a _ { 3 } = 8 , a _ { 7 } = 36a3​=8,a7​=36

Find the nth term of the geometric sequence. −2,−52,−2.58,K- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }−2,−25​,−82.5​,K

Logs are stacked so that there are 17 logs in the bottom row, 16 logs in the second row from the bottom, and so on, decreasing by 1 log each row. How many logs are there in the first five rows from the bottom?

Write the first five terms of the geometric sequence. a1=−1,r=−16a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }a1​=−1,r=−61​

Fundamental Concepts of Algebra

Equations and Inequalities

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Simplify the factorial expression. $\frac { 14 ! } { 12 ! }$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 7 } 3 \left( \frac { 2 } { 5 } \right) ^ { n - 1 }$

Write the rational number $0 . \overline { 81 }$ as the quotient of two integers in simplest form.

Write the given series in summation notation. $\frac { 1 } { 5 } + \frac { 1 } { 40 } + \frac { 1 } { 135 } + \frac { 1 } { 320 } + \frac { 1 } { 625 } + \frac { 1 } { 1080 }$

Use the Ratio Test to determine the convergence or divergence of the series. $\sum _ { n = 1 } ^ { \infty } n \left( \frac { 8 } { 5 } \right) ^ { n }$

Determine the convergence or divergence of the series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { 3 } { 10 } \right) ^ { n }$

Write the first five terms of the power series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x - 8 ) ^ { n } } { 4 ^ { n } }$ .

A ball is dropped from a height of 14 feet, and on each rebound it rises to $\frac { 2 } { 5 }$ its preceding height. Write an expression for the height of the nth rebound.

Determine the convergence or divergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { 6 ^ { n } } { 900 }$ . Use a symbolic algebra utility to verify your result.

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

Express the value of the given repeating decimal as a fraction. [Hint: Write as an infinite series.] $0 . \overline { 48 }$

Find the sum of the convergent series. $\sum _ { n = 0 } ^ { \infty } 9 \left( \frac { 8 } { 9 } \right) ^ { n }$

Write the first five terms of the sequence. an = $2 - \frac { 5 } { n } - \frac { 7 } { n ^ { 2 } }$

Determine the convergence or divergence of the following series. $\sum _ { n = 1 } ^ { \infty } \frac { 2 } { n ^ { 2 } }$

Find a formula for an for the arithmetic sequence. $a _ { 3 } = 8 , a _ { 7 } = 36$

Find the nth term of the geometric sequence. $- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }$

Write the first five terms of the geometric sequence. $a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }$