Question 1

Solve the problem.
-After being struck with a hammer, a gong vibrates 54 vibrations in the first second and in each second thereafter makes  67\frac { 6 } { 7 }76​  as many vibrations as in the previous second. Find how many vibrations the gong makes before it
Stops vibrating.

Accepted Answer

A)   388 vibrations
B)   63 vibrations
C)   58 vibrations
D)   378 vibrations

Question 2

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$3 + 3 \cdot \frac { 1 } { 7 } + 3 \cdot \left( \frac { 1 } { 7 } \right) ^ { 2 } + \ldots + 3 \cdot \left( \frac { 1 } { 7 } \right) ^ { n - 1 } = \frac { 3 \left( 1 - \left( \frac { 1 } { 7 } \right) ^ { n } \right) } { 1 - \frac { 1 } { 7 } }$$

Accepted Answer

\[\begin{array} { l }
= \frac { 3 \left...

Question 3

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

Accepted Answer

A)   4, 20, 84, 340
B)   4, 16, 64, 256
C)   4, 12, 60, 252
D)   4, 12, 44, 172

Question 4

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$n ^ { 2 } - n + 2 \text { is divisible by } 2$$

Accepted Answer

First, we show that the statement is tru...

Question 5

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
- a=2;r=3a = 2 ; r = 3a=2;r=3

Accepted Answer

A)    a5=162;an=2⋅(3) n−1a _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 )  ^ { n - 1 }a5​=162;an​=2⋅(3) n−1 
B)    a5=486;an=2⋅(3) n\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 )  ^ { \mathrm { n } }a5​=486;an​=2⋅(3) n 
C)    a5=162;an=2⋅(3) na _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 )  ^ { n }a5​=162;an​=2⋅(3) n 
D)    a5=486;an=2⋅(3) n−1\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 )  ^ { \mathrm { n } - 1 }a5​=486;an​=2⋅(3) n−1

Question 6

Solve.
-The number of students in a school in year n is estimated by the model  an=5n2+14n+81a _ { n } = 5 n ^ { 2 } + 14 n + 81an​=5n2+14n+81  . About how many students are in the school in each of the first three years?

Accepted Answer

A)   100, 129, 153
B)   100, 129, 168
C)   114, 143, 182
D)   105, 129, 168

Question 7

Find the sum of the sequence.
- ∑k=1413k\sum _ { k = 1 } ^ { 4 } \frac { 1 } { 3 k }∑k=14​3k1​

Accepted Answer

A)    1118\frac { 11 } { 18 }1811​ 
B)    512\frac { 5 } { 12 }125​ 
C)    2536\frac { 25 } { 36 }3625​ 
D)    112\frac { 1 } { 12 }121​

Question 8

Find the sum of the arithmetic sequence.
-{-6n - 1}, n = 38

Accepted Answer

A)   -4,370
B)   -4,313
C)   -4,180
D)   -4,484

Question 9

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
-6 th term is  −17;15- 17 ; 15−17;15  th term is  −53- 53−53

Accepted Answer

A)    a1=3,d=−4,an=an−1−4a _ { 1 } = 3 , d = - 4 , a _ { n } = a _ { n - 1 } - 4a1​=3,d=−4,an​=an−1​−4 
B)    a1=3,d=4,an=an−1+4a _ { 1 } = 3 , d = 4 , a _ { n } = a _ { n - 1 } + 4a1​=3,d=4,an​=an−1​+4 
C)    a1=7,d=4,an=an−1+4a _ { 1 } = 7 , d = 4 , a _ { n } = a _ { n - 1 } + 4a1​=7,d=4,an​=an−1​+4 
D)    a1=7,d=−4,an=an−1−4a _ { 1 } = 7 , d = - 4 , a _ { n } = a _ { n - 1 } - 4a1​=7,d=−4,an​=an−1​−4

Question 10

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
- ∑k=1∞−2(−0.9) k−1\sum _ { k = 1 } ^ { \infty } - 2 ( - 0.9 )  ^ { \mathrm { k } - 1 }∑k=1∞​−2(−0.9) k−1

Accepted Answer

A)   Converges; -1.05
B)   Converges; 2.65
C)   Diverges
D)   Converges; 1.05

Question 11

Find the sum of the arithmetic sequence.
-(-6)  + (-1)  + 4 + 9 + ... + 39

Accepted Answer

A)   165
B)   330
C)   170
D)   160

Question 12

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
- ∑k=1∞4(32) k−1\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 3 } { 2 } \right)  ^ { k - 1 }∑k=1∞​4(23​) k−1

Accepted Answer

A)   Converges; 12
B)   Converges; 4
C)   Diverges
D)   Converges; 16

Question 13

Evaluate the expression.
- (2633) \left( \begin{array} { c } 263 \ 3 \end{array} ight) (2633​)

Accepted Answer

A)   260
B)    263!260!\frac { 263 ! } { 260 ! }260!263!​ 
C)    2,997,4112,997,4112,997,411 
D)    17,984,46617,984,46617,984,466

Question 14

Express the repeating decimal as a fraction in lowest terms.
- 0.77‾=77100+7710,000+771,000,000+…0 . \overline { 77 } = \frac { 77 } { 100 } + \frac { 77 } { 10,000 } + \frac { 77 } { 1,000,000 } + \ldots0.77=10077​+10,00077​+1,000,00077​+…

Accepted Answer

A)    777710000\frac { 7777 } { 10000 }100007777​ 
B)    7777999\frac { 7777 } { 999 }9997777​ 
C)    79\frac { 7 } { 9 }97​ 
D)    7700999\frac { 7700 } { 999 }9997700​

Question 15

Determine whether the sequence is geometric.
-4, 12, 36, 108, 324, ...

Accepted Answer

A)   Geometric
B)   Not geometric

Question 16

Find the indicated term using the given information.
- a10=35,a18=75;a1a 10 = 35 , a 18 = 75 ; a 1a10=35,a18=75;a1

Accepted Answer

A)   5
B)   0
C)   -5
D)   -10

Question 17

Find the sum of the sequence.
- ∑k=25(4k−5) \sum _ { k = 2 } ^ { 5 } ( 4 k - 5 ) ∑k=25​(4k−5)

Accepted Answer

A)   36
B)   33
C)   23
D)   30

Question 18

Find the sum of the sequence.
- ∑k=15(k+1) \sum _ { k = 1 } ^ { 5 } ( k + 1 ) ∑k=15​(k+1)

Accepted Answer

A)   14
B)   20
C)   6
D)   8

Question 19

Expand the expression using the Binomial Theorem.
- (5x+2) 4( 5 x + 2 )  ^ { 4 }(5x+2) 4

Accepted Answer

A)    1,250x4+2,000x3+600x2+320x+161,250 x ^ { 4 } + 2,000 x ^ { 3 } + 600 x ^ { 2 } + 320 x + 161,250x4+2,000x3+600x2+320x+16 
B)    625x4+1,000x3+600x2+160x+16625 x ^ { 4 } + 1,000 x ^ { 3 } + 600 x ^ { 2 } + 160 x + 16625x4+1,000x3+600x2+160x+16 
C)    (25x2+10x+4) 4\left( 25 x ^ { 2 } + 10 x + 4 \right)  ^ { 4 }(25x2+10x+4) 4 
D)    625x3+1,000x2+600x+160625 x ^ { 3 } + 1,000 x ^ { 2 } + 600 x + 160625x3+1,000x2+600x+160

Question 20

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.
-Find ag when  a1=4,000,r=13\mathrm { a } _ { 1 } = 4,000 , \mathrm { r } = \frac { 1 } { 3 }a1​=4,000,r=31​ .

Accepted Answer

A)    400059049\frac { 4000 } { 59049 }590494000​ 
B)    120083\frac { 12008 } { 3 }312008​ 
C)    40006561\frac { 4000 } { 6561 }65614000​ 
D)    400019683\frac { 4000 } { 19683 }196834000​

Exam 11: Sequences; Induction; the Binomial Theorem

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $\sum _ { k = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }$

Correct Answer
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Solve the problem. -After being struck with a hammer, a gong vibrates 54 vibrations in the first second and in each second thereafter makes $\frac { 6 } { 7 }$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it Stops vibrating.

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 4 } \frac { 1 } { 3 k }$

Correct Answer
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Expand the expression using the Binomial Theorem. - $( 5 x + 2 ) ^ { 4 }$

Correct Answer
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Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 3 } { 2 } \right) ^ { k - 1 }$

Correct Answer
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Correct Answer
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\[\begin{array} { l }
= \frac { 3 \left...
View Answer

Evaluate the expression. - $\left( \begin{array} { c } 263 \\ 3 \end{array} \right)$

Correct Answer
verified

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. -Find ag when $\mathrm { a } _ { 1 } = 4,000 , \mathrm { r } = \frac { 1 } { 3 }$ .

Correct Answer
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Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 5 } ( 4 k - 5 )$

Correct Answer
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Find the indicated term using the given information. - $a 10 = 35 , a 18 = 75 ; a 1$

Correct Answer
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Find the sum of the arithmetic sequence. -{-6n - 1}, n = 38

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

Correct Answer
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Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 5 } ( k + 1 )$

Correct Answer
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Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $a = 2 ; r = 3$

Correct Answer
verified

Solve. -The number of students in a school in year n is estimated by the model $a _ { n } = 5 n ^ { 2 } + 14 n + 81$ . About how many students are in the school in each of the first three years?

Correct Answer
verified

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $n ^ { 2 } - n + 2 \text { is divisible by } 2$

Correct Answer
verified
First, we show that the statement is tru...
View Answer

Express the repeating decimal as a fraction in lowest terms. - $0 . \overline { 77 } = \frac { 77 } { 100 } + \frac { 77 } { 10,000 } + \frac { 77 } { 1,000,000 } + \ldots$

Correct Answer
verified

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. -6 th term is $- 17 ; 15$ th term is $- 53$

Correct Answer
verified

Determine whether the sequence is geometric. -4, 12, 36, 108, 324, ...

Correct Answer
verified

Find the sum of the arithmetic sequence. -(-6) + (-1) + 4 + 9 + ... + 39

Correct Answer
verified

Exam 11: Sequences; Induction; the Binomial Theorem

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - ∑k=1∞−2(−0.9) k−1\sum _ { k = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }∑k=1∞​−2(−0.9) k−1

Correct AnswerverifiedShow Answer

Solve the problem. -After being struck with a hammer, a gong vibrates 54 vibrations in the first second and in each second thereafter makes 67\frac { 6 } { 7 }76​ as many vibrations as in the previous second. Find how many vibrations the gong makes before it Stops vibrating.

Correct AnswerverifiedShow Answer

Find the sum of the sequence. - ∑k=1413k\sum _ { k = 1 } ^ { 4 } \frac { 1 } { 3 k }∑k=14​3k1​

Correct AnswerverifiedShow Answer

Expand the expression using the Binomial Theorem. - (5x+2) 4( 5 x + 2 ) ^ { 4 }(5x+2) 4

Correct AnswerverifiedShow Answer

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - ∑k=1∞4(32) k−1\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 3 } { 2 } \right) ^ { k - 1 }∑k=1∞​4(23​) k−1

Correct AnswerverifiedShow Answer

Correct Answerverified\[\begin{array} { l } = \frac { 3 \left...View AnswerShow Answer

Evaluate the expression. - (2633) \left( \begin{array} { c } 263 \\ 3 \end{array} \right) (2633​)

Correct AnswerverifiedShow Answer

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. -Find ag when a1=4,000,r=13\mathrm { a } _ { 1 } = 4,000 , \mathrm { r } = \frac { 1 } { 3 }a1​=4,000,r=31​ .

Correct AnswerverifiedShow Answer

Find the sum of the sequence. - ∑k=25(4k−5) \sum _ { k = 2 } ^ { 5 } ( 4 k - 5 ) ∑k=25​(4k−5)

Correct AnswerverifiedShow Answer

Find the indicated term using the given information. - a10=35,a18=75;a1a 10 = 35 , a 18 = 75 ; a 1a10=35,a18=75;a1

Correct AnswerverifiedShow Answer

Find the sum of the arithmetic sequence. -{-6n - 1}, n = 38

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Find the sum of the sequence. - ∑k=15(k+1) \sum _ { k = 1 } ^ { 5 } ( k + 1 ) ∑k=15​(k+1)

Correct AnswerverifiedShow Answer

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - a=2;r=3a = 2 ; r = 3a=2;r=3

Correct AnswerverifiedShow Answer

Solve. -The number of students in a school in year n is estimated by the model an=5n2+14n+81a _ { n } = 5 n ^ { 2 } + 14 n + 81an​=5n2+14n+81 . About how many students are in the school in each of the first three years?

Correct AnswerverifiedShow Answer

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - n2−n+2 is divisible by 2n ^ { 2 } - n + 2 \text { is divisible by } 2n2−n+2 is divisible by 2

Correct AnswerverifiedFirst, we show that the statement is tru...View AnswerShow Answer

Express the repeating decimal as a fraction in lowest terms. - 0.77‾=77100+7710,000+771,000,000+…0 . \overline { 77 } = \frac { 77 } { 100 } + \frac { 77 } { 10,000 } + \frac { 77 } { 1,000,000 } + \ldots0.77=10077​+10,00077​+1,000,00077​+…

Correct AnswerverifiedShow Answer

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. -6 th term is −17;15- 17 ; 15−17;15 th term is −53- 53−53

Correct AnswerverifiedShow Answer

Determine whether the sequence is geometric. -4, 12, 36, 108, 324, ...

Correct AnswerverifiedShow Answer

Find the sum of the arithmetic sequence. -(-6) + (-1) + 4 + 9 + ... + 39

Correct AnswerverifiedShow Answer

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $\sum _ { k = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }$

Correct Answer
verified

Solve the problem. -After being struck with a hammer, a gong vibrates 54 vibrations in the first second and in each second thereafter makes $\frac { 6 } { 7 }$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it Stops vibrating.

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 4 } \frac { 1 } { 3 k }$

Correct Answer
verified

Expand the expression using the Binomial Theorem. - $( 5 x + 2 ) ^ { 4 }$

Correct Answer
verified

Determine whether the infinite geometric series converges or diverges. If it converges, find its sum. - $\sum _ { k = 1 } ^ { \infty } 4 \left( \frac { 3 } { 2 } \right) ^ { k - 1 }$

Correct Answer
verified

Correct Answer
verified
\[\begin{array} { l }
= \frac { 3 \left...
View Answer

Evaluate the expression. - $\left( \begin{array} { c } 263 \\ 3 \end{array} \right)$

Correct Answer
verified

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. -Find ag when $\mathrm { a } _ { 1 } = 4,000 , \mathrm { r } = \frac { 1 } { 3 }$ .

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 5 } ( 4 k - 5 )$

Correct Answer
verified

Find the indicated term using the given information. - $a 10 = 35 , a 18 = 75 ; a 1$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 5 } ( k + 1 )$

Correct Answer
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Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $a = 2 ; r = 3$

Correct Answer
verified

Solve. -The number of students in a school in year n is estimated by the model $a _ { n } = 5 n ^ { 2 } + 14 n + 81$ . About how many students are in the school in each of the first three years?

Correct Answer
verified

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $n ^ { 2 } - n + 2 \text { is divisible by } 2$

Correct Answer
verified
First, we show that the statement is tru...
View Answer

Express the repeating decimal as a fraction in lowest terms. - $0 . \overline { 77 } = \frac { 77 } { 100 } + \frac { 77 } { 10,000 } + \frac { 77 } { 1,000,000 } + \ldots$

Correct Answer
verified

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. -6 th term is $- 17 ; 15$ th term is $- 53$

Correct Answer
verified

Correct Answer
verified

Correct Answer
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