Question 1

Write out the first five terms of the sequence.
- {4n2}\left\{ \frac { 4 } { n ^ { 2 } } \right\}{n24​}

Accepted Answer

A)    4,1,49,14,4254,1 , \frac { 4 } { 9 } , \frac { 1 } { 4 } , \frac { 4 } { 25 }4,1,94​,41​,254​ 
B)    14,29,316,425,536\frac { 1 } { 4 } , \frac { 2 } { 9 } , \frac { 3 } { 16 } , \frac { 4 } { 25 } , \frac { 5 } { 36 }41​,92​,163​,254​,365​ 
C)    1,12,13,14,151 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 }1,21​,31​,41​,51​ 
D)    1,14,19,116,1251 , \frac { 1 } { 4 } , \frac { 1 } { 9 } , \frac { 1 } { 16 } , \frac { 1 } { 25 }1,41​,91​,161​,251​

Question 2

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

Accepted Answer

First we show that the statement is true...

Question 3

Accepted Answer

Find the nth term of the geometric sequence.-5, -15, 45, -135, 405 
A)    an=5⋅(−3) n−1a _ { n } = 5 \cdot ( - 3 )  ^ { n - 1 }an​=5⋅(−3) n−1 
B)    an=5⋅(−3) na _ { n } = 5 \cdot ( - 3 )  nan​=5⋅(−3) n 
C)    an=a1−3na _ { n } = a _ { 1 } - 3 ^ { n }an​=a1​−3n 
D)    an=5⋅(−3) na _ { n } = 5 \cdot ( - 3 )  ^ { n }an​=5⋅(−3) n

Question 4

Determine whether the sequence is arithmetic.
-5, -15, 45, -135, 405, ...

Accepted Answer

A)   Arithmetic
B)   Not arithmetic

Question 5

Expand the expression using the Binomial Theorem.
- (w−s) 6( w - s )  ^ { 6 }(w−s) 6

Accepted Answer

A)    w6−6w5s+15w4s2−20w3s3+15w2s4−6ws5+s6w ^ { 6 } - 6 w ^ { 5 } s + 15 w ^ { 4 } s ^ { 2 } - 20 w ^ { 3 } s ^ { 3 } + 15 w ^ { 2 } s ^ { 4 } - 6 w s ^ { 5 } + s ^ { 6 }w6−6w5s+15w4s2−20w3s3+15w2s4−6ws5+s6 
B)    w6−8w5s+17w4s2−22w3s3+17w2s4−8ws5+s6w ^ { 6 } - 8 w ^ { 5 } s + 17 w ^ { 4 } s ^ { 2 } - 22 w ^ { 3 } s ^ { 3 } + 17 w ^ { 2 } s ^ { 4 } - 8 w s ^ { 5 } + s ^ { 6 }w6−8w5s+17w4s2−22w3s3+17w2s4−8ws5+s6 
C)    w6−s6w ^ { 6 } - s ^ { 6 }w6−s6 
D)    w6−6w5s−30w4s2+120w3s3+360w2s4−720ws5−720s6w ^ { 6 } - 6 w ^ { 5 } s - 30 w ^ { 4 } s ^ { 2 } + 120 w ^ { 3 } s ^ { 3 } + 360 w ^ { 2 } s ^ { 4 } - 720 w s ^ { 5 } - 720 s ^ { 6 }w6−6w5s−30w4s2+120w3s3+360w2s4−720ws5−720s6

Question 6

Accepted Answer

The given pattern continues. Write down the nth term of the sequence suggested by the pattern.-0, 2, 6, 12, 20, ... 
A)    an=4n−6a _ { n } = 4 n - 6an​=4n−6 
B)    an=2n−1−1a _ { n } = 2 ^ { n - 1 } - 1an​=2n−1−1 
C)    an=n2−na _ { n } = n ^ { 2 } - nan​=n2−n 
D)    an=2n−2a _ { n } = 2 n - 2an​=2n−2

Question 7

Find the sum of the sequence.
- ∑k=25(−1) k+1(k−9) 2\sum _ { k = 2 } ^ { 5 } ( - 1 )  ^ { k + 1 } ( k - 9 )  ^ { 2 }∑k=25​(−1) k+1(k−9) 2

Accepted Answer

A)   -1,914
B)   126
C)   -22
D)   -126

Question 8

Find the indicated term using the given information.
- a45=−1475,a13=−515;a3\mathrm { a } _ { 45 } = - \frac { 147 } { 5 } , \mathrm { a } _ { 13 } = - \frac { 51 } { 5 } ; \mathrm { a } _ { 3 }a45​=−5147​,a13​=−551​;a3​

Accepted Answer

A)    −215- \frac { 21 } { 5 }−521​ В)  15
C)    −245- \frac { 24 } { 5 }−524​ 
D)    −35- \frac { 3 } { 5 }−53​

Question 9

Find the sum.
-2, -4, 8, -16, 32

Accepted Answer

A)   -62
B)   62
C)   -22
D)   22

Question 10

The sequence is defined recursively. Write the first four terms.
- a1=3 and an=2an−1 for n≥2a _ { 1 } = 3 	ext { and } a _ { n } = 2 a _ { n - 1 } 	ext { for } n \geq 2a1​=3 and an​=2an−1​ for n≥2

Accepted Answer

A)   3, 5, 4, 3
B)   3, 6, 12, 24
C)   3, 8, 14, 26
D)   4, 8, 16, 32

Question 11

Write out the sum. Do not evaluate.
- ∑k=16(5k−3) \sum _ { k = 1 } ^ { 6 } ( 5 k - 3 ) ∑k=16​(5k−3)

Accepted Answer

A)    2+7+12+17+22+27+…2 + 7 + 12 + 17 + 22 + 27 + \ldots2+7+12+17+22+27+… 
B)    1+2+7+12+17+221 + 2 + 7 + 12 + 17 + 221+2+7+12+17+22 
C)   60
D)    2+7+12+17+22+272 + 7 + 12 + 17 + 22 + 272+7+12+17+22+27

Question 12

Find the sum of the sequence.
- ∑k=14(−k) \sum _ { k = 1 } ^ { 4 } ( - k ) ∑k=14​(−k)

Accepted Answer

A)   -12
B)   -4
C)   -3
D)   -10  D

Question 13

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so.
-4, -12, 36, -108, 324

Accepted Answer

A)   -3
B)   -16
C)   3
D)   Not geometric

Question 14

Write out the first five terms of the sequence.
- {(2n−1) !n!}\left\{ \frac { ( 2 n - 1 )  ! } { n ! } \right\}{n!(2n−1) !​}

Accepted Answer

A)    2,6,40,1260,72,5762,6,40,1260,72,5762,6,40,1260,72,576 
B)    1,6,40,1260,72,5761,6,40,1260,72,5761,6,40,1260,72,576 
C)    2,3,20,210,30242,3,20,210,30242,3,20,210,3024 
D)    1,3,20,210,30241,3,20,210,30241,3,20,210,3024 D

Question 15

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
- ∑k=153(−4) k\sum _ { k = 1 } ^ { 5 } 3 ( - 4 )  ^ { k }∑k=15​3(−4) k

Accepted Answer

A)   -2,460
B)   244
C)   -7,710
D)   -156

Question 16

Solve the problem.
-Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500
per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Accepted Answer

The answer of Solve the problem.
-Suppose you just received a...

Question 17

Solve the problem.
-Use the Binomial Theorem to approximate $( 0.98 ) ^ { 6 } = \left( 1 - 2 \left( 10 ^ { - 2 } \right) \right) ^ { 6 }$ to 5 decimal places.

Accepted Answer

0.88584

Question 18

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$$

Accepted Answer

First we show that the statement is true...

Question 19

Find the sum of the arithmetic sequence.
-{-6n + 2}, n = 36

Accepted Answer

A)   -3,636
B)   -3,924
C)   -3,762
D)   -3,816

Question 20

Accepted Answer

Solve the problem.-For the geometric sequence 2, 1,  2,1,12,14,…, find an2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , 	ext { find } a _ { n }2,1,21​,41​,…, find an​  . 
A)    an=21−na _ { n } = 2 ^ { 1 - n }an​=21−n 
B)    an=(12) 1−na _ { n } = \left( \frac { 1 } { 2 } ight)  ^ { 1 - n }an​=(21​) 1−n 
C)    an=2n−2a _ { n } = 2 ^ { n - 2 }an​=2n−2 
D)    an=(12) n−2a _ { n } = \left( \frac { 1 } { 2 } ight)  ^ { n - 2 }an​=(21​) n−2

Exam 11: Sequences; Induction; the Binomial Theorem

Write out the first five terms of the sequence. - $\left\{ \frac { 4 } { n ^ { 2 } } \right\}$

Correct Answer
verified

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

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

Correct Answer
verified

Determine whether the sequence is arithmetic. -5, -15, 45, -135, 405, ...

Correct Answer
verified

Expand the expression using the Binomial Theorem. - $( w - s ) ^ { 6 }$

Correct Answer
verified

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 5 } ( - 1 ) ^ { k + 1 } ( k - 9 ) ^ { 2 }$

Correct Answer
verified

Find the indicated term using the given information. - $\mathrm { a } _ { 45 } = - \frac { 147 } { 5 } , \mathrm { a } _ { 13 } = - \frac { 51 } { 5 } ; \mathrm { a } _ { 3 }$

Correct Answer
verified

Find the sum. -2, -4, 8, -16, 32

Correct Answer
verified

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

Correct Answer
verified

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { 6 } ( 5 k - 3 )$

Correct Answer
verified

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

Correct Answer
verified
D

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -4, -12, 36, -108, 324

Correct Answer
verified

Write out the first five terms of the sequence. - $\left\{ \frac { ( 2 n - 1 ) ! } { n ! } \right\}$

Correct Answer
verified
D

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 5 } 3 ( - 4 ) ^ { k }$

Correct Answer
verified

Solve the problem. -Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Correct Answer
verified

Solve the problem. -Use the Binomial Theorem to approximate $( 0.98 ) ^ { 6 } = \left( 1 - 2 \left( 10 ^ { - 2 } \right) \right) ^ { 6 }$ to 5 decimal places.

Correct Answer
verified
0.88584

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$

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

Find the sum of the arithmetic sequence. -{-6n + 2}, n = 36

Correct Answer
verified

Correct Answer
verified

Exam 11: Sequences; Induction; the Binomial Theorem

Write out the first five terms of the sequence. - {4n2}\left\{ \frac { 4 } { n ^ { 2 } } \right\}{n24​}

Correct AnswerverifiedShow Answer

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - 1⋅1+2⋅1+3⋅1+…+n=n(n+1)21 \cdot 1 + 2 \cdot 1 + 3 \cdot 1 + \ldots + n = \frac { n ( n + 1 ) } { 2 }1⋅1+2⋅1+3⋅1+…+n=2n(n+1)​

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

Correct AnswerverifiedShow Answer

Determine whether the sequence is arithmetic. -5, -15, 45, -135, 405, ...

Correct AnswerverifiedShow Answer

Expand the expression using the Binomial Theorem. - (w−s) 6( w - s ) ^ { 6 }(w−s) 6

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the sum of the sequence. - ∑k=25(−1) k+1(k−9) 2\sum _ { k = 2 } ^ { 5 } ( - 1 ) ^ { k + 1 } ( k - 9 ) ^ { 2 }∑k=25​(−1) k+1(k−9) 2

Correct AnswerverifiedShow Answer

Find the indicated term using the given information. - a45=−1475,a13=−515;a3\mathrm { a } _ { 45 } = - \frac { 147 } { 5 } , \mathrm { a } _ { 13 } = - \frac { 51 } { 5 } ; \mathrm { a } _ { 3 }a45​=−5147​,a13​=−551​;a3​

Correct AnswerverifiedShow Answer

Find the sum. -2, -4, 8, -16, 32

Correct AnswerverifiedShow Answer

The sequence is defined recursively. Write the first four terms. - a1=3 and an=2an−1 for n≥2a _ { 1 } = 3 \text { and } a _ { n } = 2 a _ { n - 1 } \text { for } n \geq 2a1​=3 and an​=2an−1​ for n≥2

Correct AnswerverifiedShow Answer

Write out the sum. Do not evaluate. - ∑k=16(5k−3) \sum _ { k = 1 } ^ { 6 } ( 5 k - 3 ) ∑k=16​(5k−3)

Correct AnswerverifiedShow Answer

Find the sum of the sequence. - ∑k=14(−k) \sum _ { k = 1 } ^ { 4 } ( - k ) ∑k=14​(−k)

Correct AnswerverifiedD

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. -4, -12, 36, -108, 324

Correct AnswerverifiedShow Answer

Write out the first five terms of the sequence. - {(2n−1) !n!}\left\{ \frac { ( 2 n - 1 ) ! } { n ! } \right\}{n!(2n−1) !​}

Correct AnswerverifiedD

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - ∑k=153(−4) k\sum _ { k = 1 } ^ { 5 } 3 ( - 4 ) ^ { k }∑k=15​3(−4) k

Correct AnswerverifiedShow Answer

Solve the problem. -Suppose you just received a job offer with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How many years will it be before you've made a total (or aggregate) salary of $1,025,000?

Correct AnswerverifiedShow Answer

Solve the problem. -Use the Binomial Theorem to approximate (0.98)6=(1−2(10−2))6( 0.98 ) ^ { 6 } = \left( 1 - 2 \left( 10 ^ { - 2 } \right) \right) ^ { 6 }(0.98)6=(1−2(10−2))6 to 5 decimal places.

Correct Answerverified0.88584

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - (35)n=35n\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }(35)n=35n

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

Find the sum of the arithmetic sequence. -{-6n + 2}, n = 36

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Write out the first five terms of the sequence. - $\left\{ \frac { 4 } { n ^ { 2 } } \right\}$

Correct Answer
verified

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

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

Correct Answer
verified

Correct Answer
verified

Expand the expression using the Binomial Theorem. - $( w - s ) ^ { 6 }$

Correct Answer
verified

Correct Answer
verified

Find the sum of the sequence. - $\sum _ { k = 2 } ^ { 5 } ( - 1 ) ^ { k + 1 } ( k - 9 ) ^ { 2 }$

Correct Answer
verified

Find the indicated term using the given information. - $\mathrm { a } _ { 45 } = - \frac { 147 } { 5 } , \mathrm { a } _ { 13 } = - \frac { 51 } { 5 } ; \mathrm { a } _ { 3 }$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { 6 } ( 5 k - 3 )$

Correct Answer
verified

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

Correct Answer
verified
D

Correct Answer
verified

Write out the first five terms of the sequence. - $\left\{ \frac { ( 2 n - 1 ) ! } { n ! } \right\}$

Correct Answer
verified
D

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 5 } 3 ( - 4 ) ^ { k }$

Correct Answer
verified

Correct Answer
verified

Solve the problem. -Use the Binomial Theorem to approximate $( 0.98 ) ^ { 6 } = \left( 1 - 2 \left( 10 ^ { - 2 } \right) \right) ^ { 6 }$ to 5 decimal places.

Correct Answer
verified
0.88584

Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $\left( 3 ^ { 5 } \right) ^ { n } = 3 ^ { 5 n }$

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

Correct Answer
verified

Correct Answer
verified