Question 1

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 $- 53$

Accepted Answer

A)  $a _ { 1 } = 3 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
B)  $a _ { 1 } = 3 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
C)  $a _ { 1 } = 7 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
D)  $a _ { 1 } = 7 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
A)  $a _ { 1 } = 3 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$ 
B)  $a _ { 1 } = 3 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
C)  $a _ { 1 } = 7 , d = 4 , a _ { n } = a _ { n - 1 } + 4$ 
D)  $a _ { 1 } = 7 , d = - 4 , a _ { n } = a _ { n - 1 } - 4$

Question 2

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.

Accepted Answer

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

Question 3

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

Question 4

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

Accepted Answer

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

Question 5

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 6

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

Accepted Answer

A)  $\frac { 11 } { 18 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 25 } { 36 }$ 
D)  $\frac { 1 } { 12 }$ 
A)  $\frac { 11 } { 18 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 25 } { 36 }$ 
D)  $\frac { 1 } { 12 }$

Question 7

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 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

A)  $\frac { 7777 } { 10000 }$ 
B)  $\frac { 7777 } { 999 }$ 
C)  $\frac { 7 } { 9 }$ 
D)  $\frac { 7700 } { 999 }$ 
A)  $\frac { 7777 } { 10000 }$ 
B)  $\frac { 7777 } { 999 }$ 
C)  $\frac { 7 } { 9 }$ 
D)  $\frac { 7700 } { 999 }$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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?

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

A)  $a _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 ) ^ { n - 1 }$ 
B)  $\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 ) ^ { \mathrm { n } }$ 
C)  $a _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 ) ^ { n }$ 
D)  $\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 ) ^ { \mathrm { n } - 1 }$ 
A)  $a _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 ) ^ { n - 1 }$ 
B)  $\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 ) ^ { \mathrm { n } }$ 
C)  $a _ { 5 } = 162 ; a _ { n } = 2 \cdot ( 3 ) ^ { n }$ 
D)  $\mathrm { a } _ { 5 } = 486 ; \mathrm { a } _ { \mathrm { n } } = 2 \cdot ( 3 ) ^ { \mathrm { n } - 1 }$

Question 18

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

Accepted Answer

A)  $\frac { 4000 } { 59049 }$ 
B)  $\frac { 12008 } { 3 }$ 
C)  $\frac { 4000 } { 6561 }$ 
D)  $\frac { 4000 } { 19683 }$ 
A)  $\frac { 4000 } { 59049 }$ 
B)  $\frac { 12008 } { 3 }$ 
C)  $\frac { 4000 } { 6561 }$ 
D)  $\frac { 4000 } { 19683 }$

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

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 $- 53$

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.

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

Functions and Their Graphs

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Analytic Trigonometry

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Exam 11: Sequences; Induction; the Binomial Theorem

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

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.