Question 1

An arithmetic sequence is given. Find the common difference and write out the first four terms.
-$\{ 9 - 6 n \}$

Accepted Answer

A)  $\mathrm { d } = 2 ; 4,6,8,10$ 
B)  $\mathrm { d } = - 6 ; 3 , - 3 , - 9 , - 15$ 
C)  $\mathrm { d } = - 6 ; 3 , - 1 , - 7 , - 13$ 
D)  $d = - 6 ; - 6 , - 12 , - 18 , - 24$ 
A)  $\mathrm { d } = 2 ; 4,6,8,10$ 
B)  $\mathrm { d } = - 6 ; 3 , - 3 , - 9 , - 15$ 
C)  $\mathrm { d } = - 6 ; 3 , - 1 , - 7 , - 13$ 
D)  $d = - 6 ; - 6 , - 12 , - 18 , - 24$ B

Question 2

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

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

Question 3

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
-$$a = \sqrt { 5 } ; r = \sqrt { 5 }$$

Accepted Answer

A)  $\mathrm { a } _ { 5 } = 25 \sqrt { 5 } , \mathrm { a } _ { \mathrm { n } } = 5 ^ { \mathrm { n } / 2 }$ 
B)  $\mathrm { a } _ { 5 } = 625 \sqrt { 5 } , \mathrm { a } _ { \mathrm { n } } = 5 ^ { \mathrm { n } - 1 / 2 }$ 
C)  $a _ { 5 } = 5 \sqrt { 5 } , a _ { n } = 5 ^ { n / 2 - 1 }$ 
D)  a5 $= 3,125 , a _ { n } = 5 ^ { n }$ 
A)  $\mathrm { a } _ { 5 } = 25 \sqrt { 5 } , \mathrm { a } _ { \mathrm { n } } = 5 ^ { \mathrm { n } / 2 }$ 
B)  $\mathrm { a } _ { 5 } = 625 \sqrt { 5 } , \mathrm { a } _ { \mathrm { n } } = 5 ^ { \mathrm { n } - 1 / 2 }$ 
C)  $a _ { 5 } = 5 \sqrt { 5 } , a _ { n } = 5 ^ { n / 2 - 1 }$ 
D)  a5 $= 3,125 , a _ { n } = 5 ^ { n }$ A

Question 4

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

Accepted Answer

A)  $- 2,1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
B)  $2 , - 1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
C)  $2,1 , \frac { 4 } { 5 } , \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
D)  $- 2,1 , - \frac { 4 } { 5 } , \frac { 5 } { 7 } , - \frac { 2 } { 3 }$ 
A)  $- 2,1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
B)  $2 , - 1 , \frac { 4 } { 5 } , - \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
C)  $2,1 , \frac { 4 } { 5 } , \frac { 5 } { 7 } , \frac { 2 } { 3 }$ 
D)  $- 2,1 , - \frac { 4 } { 5 } , \frac { 5 } { 7 } , - \frac { 2 } { 3 }$

Question 5

Evaluate the factorial expression.
-$\frac { 2 ! } { 4 ! }$

Accepted Answer

A)  12 
B)  $\frac { 1 } { 2 ! }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $2 !$ 
A)  12 
B)  $\frac { 1 } { 2 ! }$ 
C)  $\frac { 1 } { 12 }$ 
D)  $2 !$

Question 6

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

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 7

Expand the expression using the Binomial Theorem.
-$$( 3 x - 2 y ) ^ { 3 }$$

Accepted Answer

A)  $9 x ^ { 3 } y - 12 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
B)  $27 x ^ { 3 } - 54 x ^ { 2 } y + 36 x y ^ { 2 } - 8 y ^ { 3 }$ 
C)  $9 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
D)  $27 x ^ { 3 } - 18 x ^ { 2 } y + 12 x y ^ { 2 } - 8 y ^ { 3 }$ 
A)  $9 x ^ { 3 } y - 12 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
B)  $27 x ^ { 3 } - 54 x ^ { 2 } y + 36 x y ^ { 2 } - 8 y ^ { 3 }$ 
C)  $9 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 }$ 
D)  $27 x ^ { 3 } - 18 x ^ { 2 } y + 12 x y ^ { 2 } - 8 y ^ { 3 }$

Question 8

Write the first four terms of the sequence whose general term is given.
-$\left\{ \frac { 3 ^ { n } } { ( n + 3 ) ! } \right\}$

Accepted Answer

A)  $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 80 } , \frac { 9 } { 560 }$ 
B)  $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 40 } , \frac { 9 } { 280 }$ 
C)  $\frac { 3 } { 7 } , \frac { 9 } { 8 } , 3 , \frac { 81 } { 10 }$ 
D)  $\frac { 3 } { 4 } , \frac { 9 } { 5 } , \frac { 9 } { 2 } , \frac { 81 } { 7 }$ 
A)  $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 80 } , \frac { 9 } { 560 }$ 
B)  $\frac { 1 } { 8 } , \frac { 3 } { 40 } , \frac { 3 } { 40 } , \frac { 9 } { 280 }$ 
C)  $\frac { 3 } { 7 } , \frac { 9 } { 8 } , 3 , \frac { 81 } { 10 }$ 
D)  $\frac { 3 } { 4 } , \frac { 9 } { 5 } , \frac { 9 } { 2 } , \frac { 81 } { 7 }$

Question 9

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)  $1 / 3$ 
B)  12 
C)  3 
D)  Not geometric 
A)  $1 / 3$ 
B)  12 
C)  3 
D)  Not geometric

Question 10

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence.
-7th term is 45; 16th term is 117

Accepted Answer

A)  a1 = -3, d = 8, an = an-1 - 8 
B)  a1 = -3, d = 8, an = an-1 + 8 
C)  a1 = -11, d = 8, an = an-1 - 8 
D)  a1 = -11, d = 8, an = an-1 + 8 
A)  a1 = -3, d = 8, an = an-1 - 8 
B)  a1 = -3, d = 8, an = an-1 + 8 
C)  a1 = -11, d = 8, an = an-1 - 8 
D)  a1 = -11, d = 8, an = an-1 + 8

Question 11

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 a12 when a1 = -4, r = 2.

Accepted Answer

A)  18 
B)  -8,192 
C)  -16,384 
D)  -8,188 
A)  18 
B)  -8,192 
C)  -16,384 
D)  -8,188

Question 12

The sequence is defined recursively. Write the first four terms.
-$a _ { 1 } = 4$ and $a _ { n } = 4 a _ { n - 1 } - 4$ 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 13

Find the nth term of the geometric sequence.
-$2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$

Accepted Answer

A)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n + 1 }$ 
B)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n }$ 
A)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n + 1 }$ 
B)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 4 } \right) ^ { n - 1 }$ 
C)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 2 \cdot \left( \frac { 1 } { 2 } \right) ^ { n }$

Question 14

An arithmetic sequence is given. Find the common difference and write out the first four terms.
-$$\left\{ \frac { 1 } { 3 } + \frac { n } { 8 } \right\}$$

Accepted Answer

A)  $\mathrm { d } = \frac { 1 } { 3 } ; \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 } , \frac { 5 } { 6 }$ 
B)  $\mathrm { d } = \frac { 1 } { 3 } ; \frac { 1 } { 3 } , \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 }$ 
C)  $\mathrm { d } = \frac { 1 } { 8 } ; \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 } , \frac { 5 } { 6 }$ 
D)  $\mathrm { d } = \frac { 1 } { 8 } ; \frac { 1 } { 3 } , \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 }$ 
A)  $\mathrm { d } = \frac { 1 } { 3 } ; \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 } , \frac { 5 } { 6 }$ 
B)  $\mathrm { d } = \frac { 1 } { 3 } ; \frac { 1 } { 3 } , \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 }$ 
C)  $\mathrm { d } = \frac { 1 } { 8 } ; \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 } , \frac { 5 } { 6 }$ 
D)  $\mathrm { d } = \frac { 1 } { 8 } ; \frac { 1 } { 3 } , \frac { 11 } { 24 } , \frac { 7 } { 12 } , \frac { 17 } { 24 }$

Question 15

Evaluate the factorial expression.
-$$\frac { n ( n + 9 ) ! } { ( n + 10 ) ! }$$

Accepted Answer

A)  $\frac { \mathrm { n } } { 10 }$ 
B)  $\frac { \mathrm { n } } { \mathrm { n } + 10 }$ 
C)  $\frac { \mathrm { n } } { ( \mathrm { n } + 10 ) ! }$ 
D)  $\frac { 1 } { n + 10 }$ 
A)  $\frac { \mathrm { n } } { 10 }$ 
B)  $\frac { \mathrm { n } } { \mathrm { n } + 10 }$ 
C)  $\frac { \mathrm { n } } { ( \mathrm { n } + 10 ) ! }$ 
D)  $\frac { 1 } { n + 10 }$

Question 16

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

Accepted Answer

A)  39 
B)  30 
C)  84 
D)  120 
A)  39 
B)  30 
C)  84 
D)  120

Question 17

Express the sum using summation notation.
-$$4 ^ { 2 } + 8 ^ { 3 } + 12 ^ { 4 } + \ldots + 32 ^ { 9 }$$

Accepted Answer

A)  $\sum _ { k = 1 } ^ { 8 } 4 k ^ { 2 k - 1 }$ 
B)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k }$ 
C)  $\sum _ { k = 1 } ^ { 8 } 2 ( k - 1 ) ^ { k + 1 }$ 
D)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k + 1 }$ 
A)  $\sum _ { k = 1 } ^ { 8 } 4 k ^ { 2 k - 1 }$ 
B)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k }$ 
C)  $\sum _ { k = 1 } ^ { 8 } 2 ( k - 1 ) ^ { k + 1 }$ 
D)  $\sum _ { k = 1 } ^ { 8 } ( 4 k ) ^ { k + 1 }$

Question 18

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

Accepted Answer

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

Question 19

Find the nth term of the geometric sequence.
-$- 8 , - 16 , - 32 , - 64 , - 128$

Accepted Answer

A)  $a _ { n } = - 8 \cdot 2 ^ { n }$ 
B)  $a _ { n } = a _ { 1 } + 2 ^ { n }$ 
C)  $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$ 
D)  $a _ { n } = - 8 \cdot 2 n$ 
A)  $a _ { n } = - 8 \cdot 2 ^ { n }$ 
B)  $a _ { n } = a _ { 1 } + 2 ^ { n }$ 
C)  $a _ { n } = - 8 \cdot 2 ^ { n - 1 }$ 
D)  $a _ { n } = - 8 \cdot 2 n$

Question 20

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of
summation.
-$$7 + 11 + 15 + 19 + \ldots + 47$$

Accepted Answer

A)  $\sum _ { k = 0 } ^ { 32 } 7 + 4 k$ 
B)  $\sum _ { \mathrm { k } = 1 } ^ { 10 } 7 + 4 \mathrm { k }$ 
C)  $\sum _ { k = 1 } ^ { 10 } 7 k + 4$ 
D)  $\sum _ { \mathrm { k } = 0 } ^ { 10 } 7 + 4 \mathrm { k }$ 
A)  $\sum _ { k = 0 } ^ { 32 } 7 + 4 k$ 
B)  $\sum _ { \mathrm { k } = 1 } ^ { 10 } 7 + 4 \mathrm { k }$ 
C)  $\sum _ { k = 1 } ^ { 10 } 7 k + 4$ 
D)  $\sum _ { \mathrm { k } = 0 } ^ { 10 } 7 + 4 \mathrm { k }$

An arithmetic sequence is given. Find the common difference and write out the first four terms. - $\{ 9 - 6 n \}$

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

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - $a = \sqrt { 5 } ; r = \sqrt { 5 }$

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

Evaluate the factorial expression. - $\frac { 2 ! } { 4 ! }$

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

Expand the expression using the Binomial Theorem. - $( 3 x - 2 y ) ^ { 3 }$

Write the first four terms of the sequence whose general term is given. - $\left\{ \frac { 3 ^ { n } } { ( n + 3 ) ! } \right\}$

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

Find the first term, the common difference, and give a recursive formula for the arithmetic sequence. -7th term is 45; 16th term is 117

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 a12 when a1 = -4, r = 2.

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

Find the nth term of the geometric sequence. - $2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \frac { 1 } { 8 }$

An arithmetic sequence is given. Find the common difference and write out the first four terms. - $\left\{ \frac { 1 } { 3 } + \frac { n } { 8 } \right\}$

Evaluate the factorial expression. - $\frac { n ( n + 9 ) ! } { ( n + 10 ) ! }$

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

Express the sum using summation notation. - $4 ^ { 2 } + 8 ^ { 3 } + 12 ^ { 4 } + \ldots + 32 ^ { 9 }$

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

Find the nth term of the geometric sequence. - $- 8 , - 16 , - 32 , - 64 , - 128$

Express the sum using summation notation with a lower limit of summation not necessarily 1 and with k for the index of summation. - $7 + 11 + 15 + 19 + \ldots + 47$

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Exam 11: Sequences; Induction; the Binomial Theorem

An arithmetic sequence is given. Find the common difference and write out the first four terms. - {9−6n}\{ 9 - 6 n \}{9−6n}

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\mathrm { a } = 2 ; \mathrm { r } = 3a=2;r=3

Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given. - a=5;r=5a = \sqrt { 5 } ; r = \sqrt { 5 }a=5​;r=5​

Write out the first five terms of the sequence. - {(−1)n−1(n+12n−1)}\left\{ ( - 1 ) ^ { n - 1 } \left( \frac { n + 1 } { 2 n - 1 } \right) \right\}{(−1)n−1(2n−1n+1​)}

Evaluate the factorial expression. - 2!4!\frac { 2 ! } { 4 ! }4!2!​

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

Expand the expression using the Binomial Theorem. - (3x−2y)3( 3 x - 2 y ) ^ { 3 }(3x−2y)3

Write the first four terms of the sequence whose general term is given. - {3n(n+3)!}\left\{ \frac { 3 ^ { n } } { ( n + 3 ) ! } \right\}{(n+3)!3n​}

If the sequence is geometric, find the common ratio. If the sequence is not geometric, say so. - 4,12,36,108,3244,12,36,108,3244,12,36,108,324