Question 1

Find the partial sum of $\sum _ { k = 1 } ^ { 30 } ( k + 8 )$ .

Accepted Answer

A)  930 
B)  705 
C)  255 
D)  240 
E)  2280 
A)  930 
B)  705 
C)  255 
D)  240 
E)  2280

Question 2

Write an expression for the n th term of the sequence $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \ldots$ Assume that n begins with 1.

Accepted Answer

A)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n + 2 } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 6 ^ { n + 1 } }$ 
C)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n + 1 } }$ 
D)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 6 ^ { n } }$ 
E)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n } }$ 
A)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n + 2 } }$ 
B)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 6 ^ { n + 1 } }$ 
C)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n + 1 } }$ 
D)  $a _ { n } = \frac { ( - 1 ) ^ { n + 1 } } { 6 ^ { n } }$ 
E)  $a _ { n } = \frac { ( - 1 ) ^ { n } } { 6 ^ { n } }$

Question 3

Use Pascal s Triangle to evaluate $${ } _ { 6 } C _ { 4 }$$ .

Accepted Answer

A)  $30$ 
B)  $360$ 
C)  $15$ 
D)  $24$ 
E)  $696$ 
A)  $30$ 
B)  $360$ 
C)  $15$ 
D)  $24$ 
E)  $696$

Question 4

Find the sum. $21+9+\frac{27}{7}+\frac{81}{49}+\ldots$

Accepted Answer

A)  $\frac{441}{10}$ 
B)  $\frac{147}{4}$ 
C)  $\frac{21}{4}$ 
D)  $\frac{1323}{4}$ 
E)  $\frac{3}{4}$ 
A)  $\frac{441}{10}$ 
B)  $\frac{147}{4}$ 
C)  $\frac{21}{4}$ 
D)  $\frac{1323}{4}$ 
E)  $\frac{3}{4}$

Question 5

Find the partial sum. $\sum_{i=1}^{4} 2\left(\frac{3}{2}\right)^{i-1}$

Accepted Answer

A)  $\frac{105}{4}$ 
B)  $\frac{17}{4}$ 
C)  $\frac{65}{4}$ 
D)  $\frac{89}{4}$ 
E)  $\frac{79}{4}$ 
A)  $\frac{105}{4}$ 
B)  $\frac{17}{4}$ 
C)  $\frac{65}{4}$ 
D)  $\frac{89}{4}$ 
E)  $\frac{79}{4}$

Question 6

Use the Binomial Theorem to approximate $1.82 ^ { 8 }$ rounded to three decimal places. For example: $( 1.02 ) ^ { 10 } = ( 1 + 0.02 ) ^ { 10 } \approx 1 + 10 ( 0.02 ) + 45 ( 0.02 ) ^ { 2 }$ .

Accepted Answer

A)  $89.235$ 
B)  $120.385$ 
C)  $92.066$ 
D)  $52.31$ 
E)  $66.145$ 
A)  $89.235$ 
B)  $120.385$ 
C)  $92.066$ 
D)  $52.31$ 
E)  $66.145$

Question 7

Find the n th partial sum of the geometric sequence. Round your answer to 2 decimal places. $22,22(1.04), 22(1.04)^{2}, 22(1.04)^{3}, \ldots, \quad n=9$

Accepted Answer

A) 330.57 
B) 173.76 
C) 752.71 
D) 232.82 
E) 296.70 
A) 330.57 
B) 173.76 
C) 752.71 
D) 232.82 
E) 296.70

Question 8

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $a_{1}=3, r=\frac{7}{4}$

Accepted Answer

A)  $a_{n}=3\left(\frac{7}{4}\right)^{n-1}$ 
B)  $a_{n}=\left(\frac{21}{4}\right)^{n-1}$ 
C)  $a_{n}=\left(\frac{7}{4}\right)^{n}$ 
D)  $a_{n}=\left(\frac{21}{4}\right)^{n}$ 
E)  $a_{n}=3\left(\frac{7}{4}\right)^{n}$ 
A)  $a_{n}=3\left(\frac{7}{4}\right)^{n-1}$ 
B)  $a_{n}=\left(\frac{21}{4}\right)^{n-1}$ 
C)  $a_{n}=\left(\frac{7}{4}\right)^{n}$ 
D)  $a_{n}=\left(\frac{21}{4}\right)^{n}$ 
E)  $a_{n}=3\left(\frac{7}{4}\right)^{n}$

Question 9

A deposit of $\$ 3,000$ is made in an account that earns $4 \%$ interest compounded yearly. The balance in the account after N years is given by $A _ { N } = 3,000 ( 1 + 0.04 ) ^ { N }$ , $N = 1,2,3 , \ldots$ Find the balance in this account after $20$ years by computing $A _ { 20 }$ . Round your answer to the nearest cent.

Accepted Answer

A)  $A _ { 20 } = \$ 5,403$ 
B)  $A _ { 20 } = \$ 62,400$ 
C)  $A _ { 20 } = \$ 6,573$ 
D)  $A _ { 20 } = \$ 3,245$ 
E)  $A _ { 20 } = \$ 6,321$ 
A)  $A _ { 20 } = \$ 5,403$ 
B)  $A _ { 20 } = \$ 62,400$ 
C)  $A _ { 20 } = \$ 6,573$ 
D)  $A _ { 20 } = \$ 3,245$ 
E)  $A _ { 20 } = \$ 6,321$

Question 10

Find the partial sum $\sum _ { i = 0 } ^ { 5 } 6 i + 2$ .

Accepted Answer

A)  104 
B)  140 
C)  102 
D)  142 
E)  120 
A)  104 
B)  140 
C)  102 
D)  142 
E)  120

Question 11

Evaluate the binomial coefficient $${ } _ { 20 } C _ { 20 }$$ .

Accepted Answer

A) 1 
B)  $0$ 
C)  $19$ 
D)  $20$ 
E)  $2$ 
A) 1 
B)  $0$ 
C)  $19$ 
D)  $20$ 
E)  $2$

Question 12

Find $a _ { 8 }$ of the sequence $a _ { n } = \frac { n ^ { 2 } - 7 } { ( n - 2 ) ! }$ .

Accepted Answer

A)  $a _ { 8 } = \frac { 32 } { 361 }$ 
B)  $a _ { 8 } = - \frac { 32 } { 361 }$ 
C)  $a _ { 8 } = - \frac { 19 } { 1680 }$ 
D)  $a _ { 8 } = \frac { 19 } { 1680 }$ 
E)  $a _ { 8 } = \frac { 19 } { 240 }$ 
A)  $a _ { 8 } = \frac { 32 } { 361 }$ 
B)  $a _ { 8 } = - \frac { 32 } { 361 }$ 
C)  $a _ { 8 } = - \frac { 19 } { 1680 }$ 
D)  $a _ { 8 } = \frac { 19 } { 1680 }$ 
E)  $a _ { 8 } = \frac { 19 } { 240 }$

Question 13

Write an expression for the n th term of the sequence $1,3 , \frac { 3 ^ { 2 } } { 2 } , \frac { 3 ^ { 3 } } { 6 } , \frac { 3 ^ { 4 } } { 24 } , \frac { 3 ^ { 5 } } { 120 } , \ldots$ Assume that n begins with 1.

Accepted Answer

A)  $a _ { n } = \frac { 3 ^ { n - 1 } } { n ! }$ 
B)  $a _ { n } = \frac { 3 ^ { n + 1 } } { ( n - 1 ) ! }$ 
C)  $\alpha _ { n } = \frac { 3 ^ { n } } { ( n - 1 ) ! }$ 
D)  $a _ { n } = \frac { 3 ^ { n } } { n ! }$ 
E)  $\alpha _ { n } = \frac { 3 ^ { n - 1 } } { ( n - 1 ) ! }$ 
A)  $a _ { n } = \frac { 3 ^ { n - 1 } } { n ! }$ 
B)  $a _ { n } = \frac { 3 ^ { n + 1 } } { ( n - 1 ) ! }$ 
C)  $\alpha _ { n } = \frac { 3 ^ { n } } { ( n - 1 ) ! }$ 
D)  $a _ { n } = \frac { 3 ^ { n } } { n ! }$ 
E)  $\alpha _ { n } = \frac { 3 ^ { n - 1 } } { ( n - 1 ) ! }$

Question 14

Find the n th partial sum of the arithmetic sequence. $8,13,18,23,28 , \ldots , n = 12$

Accepted Answer

A)  390 
B)  438 
C)  426 
D)  354 
E)  402 
A)  390 
B)  438 
C)  426 
D)  354 
E)  402

Question 15

Find the sum of the first 95 positive integers.

Accepted Answer

A)  4,370 
B)  4,560 
C)  9,025 
D)  4,655 
E)  9,120 
A)  4,370 
B)  4,560 
C)  9,025 
D)  4,655 
E)  9,120

Question 16

Evaluate the binomial coefficient $${ } _ { 7 } C _ { 0 }$$ .

Accepted Answer

A)  $0$ 
B)  $7$ 
C)  $6$ 
D)  $1$ 
E)  5 
A)  $0$ 
B)  $7$ 
C)  $6$ 
D)  $1$ 
E)  5

Question 17

A city of 700,000 people is growing at the rate of 2% per year. That is, at the end of each year, the population is 1.02 times the population at the beginning of the year. Estimate the population years 19 from now. Round to the nearest integer.

Accepted Answer

A) 1,243,091 
B) 1,019,768 
C) 1,103,829 
D) 1,082,186 
E) 1,194,821 
A) 1,243,091 
B) 1,019,768 
C) 1,103,829 
D) 1,082,186 
E) 1,194,821

Question 18

Find the common ratio of the geometric sequence. $\frac{1}{8 \pi}, \frac{1}{(8 \pi)^{2}}, \frac{1}{(8 \pi)^{3}}, \frac{1}{(8 \pi)^{4}},....$

Accepted Answer

A)  $\frac{1}{\pi^{2}}$ 
B)  $\frac{1}{\pi}$ 
C)  $\frac{1}{8 \pi}$ 
D)  $\frac{1}{8}$ 
E)  $\frac { 1 } { ( 8 \pi ) ^ { 2 } }$ 
A)  $\frac{1}{\pi^{2}}$ 
B)  $\frac{1}{\pi}$ 
C)  $\frac{1}{8 \pi}$ 
D)  $\frac{1}{8}$ 
E)  $\frac { 1 } { ( 8 \pi ) ^ { 2 } }$

Question 19

Write the first five terms of the sequence $a _ { n } = - \frac { 3 } { n + 4 }$ . Assume that n begins with 1.

Accepted Answer

A)  $a _ { 1 } = \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = \frac { 1 } { 3 }$ 
B)  $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = - \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$ 
C)  $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$ 
D)  $a _ { 1 } = - \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = - \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = - \frac { 3 } { 8 }$ 
E)  $a _ { 1 } = - \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = - \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = - \frac { 1 } { 3 }$ 
A)  $a _ { 1 } = \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = \frac { 1 } { 3 }$ 
B)  $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = - \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$ 
C)  $a _ { 1 } = \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = \frac { 3 } { 8 }$ 
D)  $a _ { 1 } = - \frac { 3 } { 4 } , a _ { 2 } = \frac { 3 } { 5 } a _ { 3 } = - \frac { 1 } { 2 } , a _ { 4 } = \frac { 3 } { 7 } , a _ { 5 } = - \frac { 3 } { 8 }$ 
E)  $a _ { 1 } = - \frac { 3 } { 5 } , a _ { 2 } = - \frac { 1 } { 2 } , a _ { 3 } = - \frac { 3 } { 7 } , a _ { 4 } = - \frac { 3 } { 8 } , a _ { 5 } = - \frac { 1 } { 3 }$

Question 20

Use Pascal s Triangle to evaluate $${ } _ { 9 } C _ { 7 }$$ .

Accepted Answer

A)  $116$ 
B)  $43$ 
C)  $99$ 
D)  $45$ 
E)  $36$ 
A)  $116$ 
B)  $43$ 
C)  $99$ 
D)  $45$ 
E)  $36$

Find the partial sum of $\sum _ { k = 1 } ^ { 30 } ( k + 8 )$ .

Write an expression for the n th term of the sequence $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \ldots$ Assume that n begins with 1.

Use Pascal s Triangle to evaluate ${ } _ { 6 } C _ { 4 }$ .

Find the sum. $21+9+\frac{27}{7}+\frac{81}{49}+\ldots$

Find the partial sum. $\sum_{i=1}^{4} 2\left(\frac{3}{2}\right)^{i-1}$

Use the Binomial Theorem to approximate $1.82 ^ { 8 }$ rounded to three decimal places. For example: $( 1.02 ) ^ { 10 } = ( 1 + 0.02 ) ^ { 10 } \approx 1 + 10 ( 0.02 ) + 45 ( 0.02 ) ^ { 2 }$ .

Find the n th partial sum of the geometric sequence. Round your answer to 2 decimal places. $22,22(1.04), 22(1.04)^{2}, 22(1.04)^{3}, \ldots, \quad n=9$

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $a_{1}=3, r=\frac{7}{4}$

Find the partial sum $\sum _ { i = 0 } ^ { 5 } 6 i + 2$ .

Evaluate the binomial coefficient ${ } _ { 20 } C _ { 20 }$ .

Find $a _ { 8 }$ of the sequence $a _ { n } = \frac { n ^ { 2 } - 7 } { ( n - 2 ) ! }$ .

Write an expression for the n th term of the sequence $1,3 , \frac { 3 ^ { 2 } } { 2 } , \frac { 3 ^ { 3 } } { 6 } , \frac { 3 ^ { 4 } } { 24 } , \frac { 3 ^ { 5 } } { 120 } , \ldots$ Assume that n begins with 1.

Find the n th partial sum of the arithmetic sequence. $8,13,18,23,28 , \ldots , n = 12$

Find the sum of the first 95 positive integers.

Evaluate the binomial coefficient ${ } _ { 7 } C _ { 0 }$ .

A city of 700,000 people is growing at the rate of 2% per year. That is, at the end of each year, the population is 1.02 times the population at the beginning of the year. Estimate the population years 19 from now. Round to the nearest integer.

Find the common ratio of the geometric sequence. $\frac{1}{8 \pi}, \frac{1}{(8 \pi)^{2}}, \frac{1}{(8 \pi)^{3}}, \frac{1}{(8 \pi)^{4}},....$

Write the first five terms of the sequence $a _ { n } = - \frac { 3 } { n + 4 }$ . Assume that n begins with 1.

Use Pascal s Triangle to evaluate ${ } _ { 9 } C _ { 7 }$ .

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Exam 11: Sequences, Series, and the Binomial Theorem

Find the partial sum of ∑k=130(k+8)\sum _ { k = 1 } ^ { 30 } ( k + 8 )∑k=130​(k+8) .

Write an expression for the n th term of the sequence 16,−136,1216,−11296,…\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \ldots61​,−361​,2161​,−12961​,… Assume that n begins with 1.

Use Pascal s Triangle to evaluate 6C4{ } _ { 6 } C _ { 4 }6​C4​ .

Find the sum. 21+9+277+8149+…21+9+\frac{27}{7}+\frac{81}{49}+\ldots21+9+727​+4981​+…

Find the partial sum. ∑i=142(32)i−1\sum_{i=1}^{4} 2\left(\frac{3}{2}\right)^{i-1}∑i=14​2(23​)i−1

Use the Binomial Theorem to approximate 1.8281.82 ^ { 8 }1.828 rounded to three decimal places. For example: (1.02)10=(1+0.02)10≈1+10(0.02)+45(0.02)2( 1.02 ) ^ { 10 } = ( 1 + 0.02 ) ^ { 10 } \approx 1 + 10 ( 0.02 ) + 45 ( 0.02 ) ^ { 2 }(1.02)10=(1+0.02)10≈1+10(0.02)+45(0.02)2 .

Find the n th partial sum of the geometric sequence. Round your answer to 2 decimal places. 22,22(1.04),22(1.04)2,22(1.04)3,…,n=922,22(1.04), 22(1.04)^{2}, 22(1.04)^{3}, \ldots, \quad n=922,22(1.04),22(1.04)2,22(1.04)3,…,n=9

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. a1=3,r=74a_{1}=3, r=\frac{7}{4}a1​=3,r=47​

Find the partial sum ∑i=056i+2\sum _ { i = 0 } ^ { 5 } 6 i + 2∑i=05​6i+2 .

Evaluate the binomial coefficient 20C20{ } _ { 20 } C _ { 20 }20​C20​ .

Find a8a _ { 8 }a8​ of the sequence an=n2−7(n−2)!a _ { n } = \frac { n ^ { 2 } - 7 } { ( n - 2 ) ! }an​=(n−2)!n2−7​ .

Write an expression for the n th term of the sequence 1,3,322,336,3424,35120,…1,3 , \frac { 3 ^ { 2 } } { 2 } , \frac { 3 ^ { 3 } } { 6 } , \frac { 3 ^ { 4 } } { 24 } , \frac { 3 ^ { 5 } } { 120 } , \ldots1,3,232​,633​,2434​,12035​,… Assume that n begins with 1.

Find the n th partial sum of the arithmetic sequence. 8,13,18,23,28,…,n=128,13,18,23,28 , \ldots , n = 128,13,18,23,28,…,n=12

Find the sum of the first 95 positive integers.

Evaluate the binomial coefficient 7C0{ } _ { 7 } C _ { 0 }7​C0​ .

A city of 700,000 people is growing at the rate of 2% per year. That is, at the end of each year, the population is 1.02 times the population at the beginning of the year. Estimate the population years 19 from now. Round to the nearest integer.

Find the common ratio of the geometric sequence. 18π,1(8π)2,1(8π)3,1(8π)4,....\frac{1}{8 \pi}, \frac{1}{(8 \pi)^{2}}, \frac{1}{(8 \pi)^{3}}, \frac{1}{(8 \pi)^{4}},....8π1​,(8π)21​,(8π)31​,(8π)41​,....

Write the first five terms of the sequence an=−3n+4a _ { n } = - \frac { 3 } { n + 4 }an​=−n+43​ . Assume that n begins with 1.

Use Pascal s Triangle to evaluate 9C7{ } _ { 9 } C _ { 7 }9​C7​ .

Fundamentals of Algebra

Linear Equations and Inequalities

Graphs and Functions

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Find the partial sum of $\sum _ { k = 1 } ^ { 30 } ( k + 8 )$ .

Write an expression for the n th term of the sequence $\frac { 1 } { 6 } , - \frac { 1 } { 36 } , \frac { 1 } { 216 } , - \frac { 1 } { 1296 } , \ldots$ Assume that n begins with 1.

Use Pascal s Triangle to evaluate ${ } _ { 6 } C _ { 4 }$ .

Find the sum. $21+9+\frac{27}{7}+\frac{81}{49}+\ldots$

Find the partial sum. $\sum_{i=1}^{4} 2\left(\frac{3}{2}\right)^{i-1}$

Use the Binomial Theorem to approximate $1.82 ^ { 8 }$ rounded to three decimal places. For example: $( 1.02 ) ^ { 10 } = ( 1 + 0.02 ) ^ { 10 } \approx 1 + 10 ( 0.02 ) + 45 ( 0.02 ) ^ { 2 }$ .

Find the n th partial sum of the geometric sequence. Round your answer to 2 decimal places. $22,22(1.04), 22(1.04)^{2}, 22(1.04)^{3}, \ldots, \quad n=9$

Find a formula for the n th term of the geometric sequence. Assume that n begins with 1. $a_{1}=3, r=\frac{7}{4}$

Find the partial sum $\sum _ { i = 0 } ^ { 5 } 6 i + 2$ .

Evaluate the binomial coefficient ${ } _ { 20 } C _ { 20 }$ .

Find $a _ { 8 }$ of the sequence $a _ { n } = \frac { n ^ { 2 } - 7 } { ( n - 2 ) ! }$ .

Write an expression for the n th term of the sequence $1,3 , \frac { 3 ^ { 2 } } { 2 } , \frac { 3 ^ { 3 } } { 6 } , \frac { 3 ^ { 4 } } { 24 } , \frac { 3 ^ { 5 } } { 120 } , \ldots$ Assume that n begins with 1.

Find the n th partial sum of the arithmetic sequence. $8,13,18,23,28 , \ldots , n = 12$

Evaluate the binomial coefficient ${ } _ { 7 } C _ { 0 }$ .

Find the common ratio of the geometric sequence. $\frac{1}{8 \pi}, \frac{1}{(8 \pi)^{2}}, \frac{1}{(8 \pi)^{3}}, \frac{1}{(8 \pi)^{4}},....$

Write the first five terms of the sequence $a _ { n } = - \frac { 3 } { n + 4 }$ . Assume that n begins with 1.

Use Pascal s Triangle to evaluate ${ } _ { 9 } C _ { 7 }$ .