Question 1

Select the first five terms of the sequence defined recursively.​ $$a _ { 1 } = 54 , a _ { k + 1 } = a _ { k } - 2$$ ​

Accepted Answer

A)  $56,58,60,62,64$ ​ 
B)  $52,58,50,62,64$ 
C)  $56,52,60,48,64$ 
D)  $54,58,50,62,46$ 
E)  $54,52,50,48,46$ 
A)  $56,58,60,62,64$ ​ 
B)  $52,58,50,62,64$ 
C)  $56,52,60,48,64$ 
D)  $54,58,50,62,46$ 
E)  $54,52,50,48,46$

Question 2

Write the first six terms of the sequence defined by the function. f(n)= 3n(n - 1)

Accepted Answer

A) 0,6,18,36,60,90 
B) -6,0,18,36,60,90 
C) 6,18,36,60,90,252 
D) -6,-18,36,-60,-90,0 
E) 6,18,36,60,90,126 
A) 0,6,18,36,60,90 
B) -6,0,18,36,60,90 
C) 6,18,36,60,90,252 
D) -6,-18,36,-60,-90,0 
E) 6,18,36,60,90,126

Question 3

Find the indicated partial sum of the series. $$\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$$ fourth partial sum

Accepted Answer

A)  $$\frac { 2 } { 81 }$$ 
B)  $$\frac { 242 } { 81 }$$ 
C)  $\frac { 80 } { 81 }$ ​ 
D)  $\frac { 26 } { 81 }$ ​ 
E)  $\frac { 82 } { 81 }$ 
A)  $$\frac { 2 } { 81 }$$ 
B)  $$\frac { 242 } { 81 }$$ 
C)  $\frac { 80 } { 81 }$ ​ 
D)  $\frac { 26 } { 81 }$ ​ 
E)  $\frac { 82 } { 81 }$

Question 4

Evaluate the sum.​ $$\sum _ { k = 1 } ^ { 6 } 5 k$$ ​

Accepted Answer

A) 105 
B) 97 
C) 30 
D) 28 
E) 210 
A) 105 
B) 97 
C) 30 
D) 28 
E) 210

Question 5

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) $$a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5$$

Accepted Answer

A)  $$a _ { n } = 29 - 5 n ; 19,14,9,4 , - 6$$ 
B)  $a _ { n } = 19 ; 19,19,19,19,19$ 
C)  $$a _ { n } = 19 ( n - 1 ) ; 14,9,4 , - 1 , - 6$$ 
D)  $a _ { n } = 19 - 5 n ; 19,14,9,4 , - 1$ 
E)  $a _ { n } = 24 - 5 n ; 19,14,9,4 , - 1$ 
A)  $$a _ { n } = 29 - 5 n ; 19,14,9,4 , - 6$$ 
B)  $a _ { n } = 19 ; 19,19,19,19,19$ 
C)  $$a _ { n } = 19 ( n - 1 ) ; 14,9,4 , - 1 , - 6$$ 
D)  $a _ { n } = 19 - 5 n ; 19,14,9,4 , - 1$ 
E)  $a _ { n } = 24 - 5 n ; 19,14,9,4 , - 1$

Question 6

Use a calculator to find the sum.Round to four decimal places. $$\sum _ { k = 4 } ^ { 12 } \frac { - 1 } { k + 2 }$$

Accepted Answer

A) -0.9682 
B) -2.2516 
C) -0.0714 
D) -0.8968 
E) -1.7516 
A) -0.9682 
B) -2.2516 
C) -0.0714 
D) -0.8968 
E) -1.7516

Question 7

Use sigma notation to write the sum. $$\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }$$

Accepted Answer

A)  $$\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 2 ) ! }$$ 
B)  $\sum _ { n = 0 } ^ { 5 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​ 
C)  $$\sum _ { n = 0 } ^ { 6 } \frac { 1 } { n ( n + 1 ) }$$ ​ 
D)  $\sum _ { n = 1 } ^ { 4 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​ 
E)  $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​ 
A)  $$\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 2 ) ! }$$ 
B)  $\sum _ { n = 0 } ^ { 5 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​ 
C)  $$\sum _ { n = 0 } ^ { 6 } \frac { 1 } { n ( n + 1 ) }$$ ​ 
D)  $\sum _ { n = 1 } ^ { 4 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​ 
E)  $\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 ) ( n + 2 ) }$ ​

Question 8

Select the first five terms of the sequence.(Assume that n begins with 1. )​ $$a _ { n } = \frac { 8 } { 5 }$$ ​

Accepted Answer

A)  $- \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 }$ ​ 
B)  $- \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 }$ ​ 
C)  $\frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 }$ ​ 
D) 0,1,2,3,4 
E)  $\frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 }$ ​ 
A)  $- \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 }$ ​ 
B)  $- \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 }$ ​ 
C)  $\frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 }$ ​ 
D) 0,1,2,3,4 
E)  $\frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 }$ ​

Question 9

Assume that the sequence is defined recursively.Find the first three terms of the sequence.​ $$a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }$$ ​

Accepted Answer

A) 36 
B) 5184 
C) 3 
D) 4 
E) 32 
A) 36 
B) 5184 
C) 3 
D) 4 
E) 32

Question 10

Select the first five terms of the sequence defined recursively.​ $$a _ { 1 } = 23 , a _ { k + 1 } = a _ { k } + 3$$ ​

Accepted Answer

A)  $$23,26,29,32,35$$ ​ 
B)  $$26,17,29,11,8$$ ​ 
C)  $20,26,14,32,8$ ​ 
D)  $20,17,14,11,8$ ​ 
E)  $23,17,29,11,35$ ​ 
A)  $$23,26,29,32,35$$ ​ 
B)  $$26,17,29,11,8$$ ​ 
C)  $20,26,14,32,8$ ​ 
D)  $20,17,14,11,8$ ​ 
E)  $23,17,29,11,35$ ​

Question 11

Find the sum. $$\sum _ { i = 1 } ^ { 5 } - 3 i + 4$$

Accepted Answer

A) -14 
B) -21 
C) -25 
D) -11 
E) -45 
A) -14 
B) -21 
C) -25 
D) -11 
E) -45

Question 12

Find the next term of the sequence. 64,125,216,343,...

Accepted Answer

A) 524 
B) 4,096 
C) 512 
D) 576 
E) none of these 
A) 524 
B) 4,096 
C) 512 
D) 576 
E) none of these

Question 13

Select the first five terms of the sequence.(Assume that n begins with 1. )​ $$a _ { n } = ( - 1 ) ^ { n } \left( \frac { n } { n + 11 } \right)$$ ​

Accepted Answer

A)  $\frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , \frac { 5 } { 16 }$ ​ 
B)  $- \frac { 1 } { 12 } , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , - \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
C)  $- \frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
D)  $- 12 , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
E)  $- \frac { 1 } { 12 } , \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
A)  $\frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , \frac { 5 } { 16 }$ ​ 
B)  $- \frac { 1 } { 12 } , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , - \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
C)  $- \frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
D)  $- 12 , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​ 
E)  $- \frac { 1 } { 12 } , \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }$ ​

Question 14

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. )​ $$1,7,13,19,25 , \ldots$$ ​

Accepted Answer

A)  $a _ { n } = - 6 n - 5$ 
B)  $a _ { n } = 6 n - 5$ 
C)  $a _ { n } = 5 n + 6$ 
D)  $a _ { n } = 5 n - 6$ 
E)  $a _ { n } = 6 n + 5$ 
A)  $a _ { n } = - 6 n - 5$ 
B)  $a _ { n } = 6 n - 5$ 
C)  $a _ { n } = 5 n + 6$ 
D)  $a _ { n } = 5 n - 6$ 
E)  $a _ { n } = 6 n + 5$

Question 15

Select the first five terms of the sequence.(Assume that n begins with 1. )​ $$a _ { n } = \left( \frac { 1 } { 5 } \right) ^ { n }$$ ​

Accepted Answer

A)  $$\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 125 } , \frac { 1 } { 25 }$$ 
B)  $$\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 125 } , \frac { 1 } { 20 } , \frac { 1 } { 3125 }$$ 
C)  $\frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 } , \frac { 1 } { 3125 }$ 
D)  $\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 20 } , \frac { 1 } { 25 }$ 
E)  $1 , \frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 }$ 
A)  $$\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 125 } , \frac { 1 } { 25 }$$ 
B)  $$\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 125 } , \frac { 1 } { 20 } , \frac { 1 } { 3125 }$$ 
C)  $\frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 } , \frac { 1 } { 3125 }$ 
D)  $\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 20 } , \frac { 1 } { 25 }$ 
E)  $1 , \frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 }$

Question 16

​Simplify the factorial expression. $$\frac { 9 ! } { 6 ! }$$

Accepted Answer

A) ​5040 
B) ​504 
C) ​3024 
D) ​72 
E) ​ $\frac { 3 } { 2 }$ 
A) ​5040 
B) ​504 
C) ​3024 
D) ​72 
E) ​ $\frac { 3 } { 2 }$

Question 17

Use sigma notation to write the sum.​ $$\frac { 1 } { 3 ( 1 ) } + \frac { 1 } { 3 ( 2 ) } + \frac { 1 } { 3 ( 3 ) } + \ldots + \frac { 1 } { 3 ( 9 ) }$$ ​

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 9 } - \frac { 1 } { 3 i }$ 
B)  $\sum _ { i = 0 } ^ { 10 } \frac { 1 } { 3 i }$ 
C)  $\sum _ { i = 1 } ^ { 9 } 3 i$ 
D)  $\sum _ { i = 1 } ^ { 9 } \frac { 1 } { 3 i }$ 
E)  $\sum _ { i = 1 } ^ { - 9 } \frac { 1 } { 3 i }$ 
A)  $\sum _ { i = 1 } ^ { 9 } - \frac { 1 } { 3 i }$ 
B)  $\sum _ { i = 0 } ^ { 10 } \frac { 1 } { 3 i }$ 
C)  $\sum _ { i = 1 } ^ { 9 } 3 i$ 
D)  $\sum _ { i = 1 } ^ { 9 } \frac { 1 } { 3 i }$ 
E)  $\sum _ { i = 1 } ^ { - 9 } \frac { 1 } { 3 i }$

Question 18

Find the indicated term of the sequence. $a _ { n } = ( - 1 ) ^ { n } ( 4 n - 2 )$ $a _ { 27 } =$ ?

Accepted Answer

A) 102 
B) -106 
C) -110 
D) 6 
E) -104 
A) 102 
B) -106 
C) -110 
D) 6 
E) -104

Question 19

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. )​ $$\frac { 2 } { 7 } , \frac { 3 } { 15 } , \frac { 4 } { 23 } , \frac { 5 } { 31 } , \frac { 6 } { 39 } , \ldots$$ ​

Accepted Answer

A)  $a _ { n } = \frac { n + 1 } { 8 n - 8 }$ 
B)  $a _ { n } = \frac { n - 1 } { 8 n + 1 }$ 
C)  $a _ { n } = \frac { n + 1 } { 8 n + 1 }$ 
D)  $a _ { n } = \frac { n - 1 } { 8 n - 1 }$ 
E)  $a _ { n } = \frac { n + 1 } { 8 n - 1 }$ 
A)  $a _ { n } = \frac { n + 1 } { 8 n - 8 }$ 
B)  $a _ { n } = \frac { n - 1 } { 8 n + 1 }$ 
C)  $a _ { n } = \frac { n + 1 } { 8 n + 1 }$ 
D)  $a _ { n } = \frac { n - 1 } { 8 n - 1 }$ 
E)  $a _ { n } = \frac { n + 1 } { 8 n - 1 }$

Question 20

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) $$a _ { 1 } = - 8 , a _ { k + 1 } = - 3 a _ { k }$$

Accepted Answer

A)  $$a _ { k } = ( 24 ) ^ { n } ; - 8 , - 512,4096 , - 32768,262144$$ ​ 
B)  $a _ { n } = - 8 ( - 3 ) ^ { n - 1 } ; - 8,24 , - 72,216 , - 648$ ​ 
C)  $$a _ { n } = - 8 n ; - 8 , - 16 , - 24 , - 32 , - 40$$ ​ 
D)  $a _ { k } = ( 24 ) ^ { n - 1 } ; - 8 , - 512,4096 , - 32768,262144$ ​ 
E)  $a _ { n } = - 8 ( - 3 ) ^ { n } ; 24 , - 72,216 , - 648,1944$ ​ 
A)  $$a _ { k } = ( 24 ) ^ { n } ; - 8 , - 512,4096 , - 32768,262144$$ ​ 
B)  $a _ { n } = - 8 ( - 3 ) ^ { n - 1 } ; - 8,24 , - 72,216 , - 648$ ​ 
C)  $$a _ { n } = - 8 n ; - 8 , - 16 , - 24 , - 32 , - 40$$ ​ 
D)  $a _ { k } = ( 24 ) ^ { n - 1 } ; - 8 , - 512,4096 , - 32768,262144$ ​ 
E)  $a _ { n } = - 8 ( - 3 ) ^ { n } ; 24 , - 72,216 , - 648,1944$ ​

Exam 52: Sequences and Series

Select the first five terms of the sequence defined recursively.​ a1=54,ak+1=ak−2a _ { 1 } = 54 , a _ { k + 1 } = a _ { k } - 2a1​=54,ak+1​=ak​−2 ​

Write the first six terms of the sequence defined by the function. f(n)= 3n(n - 1)

Find the indicated partial sum of the series. ∑i=1∞2(13)i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }∑i=1∞​2(31​)i fourth partial sum

Evaluate the sum.​ ∑k=165k\sum _ { k = 1 } ^ { 6 } 5 k∑k=16​5k ​

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) a1=19,ak+1=ak−5a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5a1​=19,ak+1​=ak​−5

Use a calculator to find the sum.Round to four decimal places. ∑k=412−1k+2\sum _ { k = 4 } ^ { 12 } \frac { - 1 } { k + 2 }∑k=412​k+2−1​

Use sigma notation to write the sum. 13⋅2+14⋅3+…+18⋅7\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }3⋅21​+4⋅31​+…+8⋅71​

Select the first five terms of the sequence.(Assume that n begins with 1. )​ an=85a _ { n } = \frac { 8 } { 5 }an​=58​ ​

Assume that the sequence is defined recursively.Find the first three terms of the sequence.​ a1=3 and an+1=4an2a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }a1​=3 and an+1​=4an2​ ​

Select the first five terms of the sequence defined recursively.​ a1=23,ak+1=ak+3a _ { 1 } = 23 , a _ { k + 1 } = a _ { k } + 3a1​=23,ak+1​=ak​+3 ​

Find the sum. ∑i=15−3i+4\sum _ { i = 1 } ^ { 5 } - 3 i + 4∑i=15​−3i+4

Find the next term of the sequence. 64,125,216,343,...

Select the first five terms of the sequence.(Assume that n begins with 1. )​ an=(−1)n(nn+11)a _ { n } = ( - 1 ) ^ { n } \left( \frac { n } { n + 11 } \right)an​=(−1)n(n+11n​) ​

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. )​ 1,7,13,19,25,…1,7,13,19,25 , \ldots1,7,13,19,25,… ​

Select the first five terms of the sequence.(Assume that n begins with 1. )​ an=(15)na _ { n } = \left( \frac { 1 } { 5 } \right) ^ { n }an​=(51​)n ​

​Simplify the factorial expression. 9!6!\frac { 9 ! } { 6 ! }6!9!​

Use sigma notation to write the sum.​ 13(1)+13(2)+13(3)+…+13(9)\frac { 1 } { 3 ( 1 ) } + \frac { 1 } { 3 ( 2 ) } + \frac { 1 } { 3 ( 3 ) } + \ldots + \frac { 1 } { 3 ( 9 ) }3(1)1​+3(2)1​+3(3)1​+…+3(9)1​ ​

Find the indicated term of the sequence. an=(−1)n(4n−2)a _ { n } = ( - 1 ) ^ { n } ( 4 n - 2 )an​=(−1)n(4n−2) a27=a _ { 27 } =a27​= ?

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. )​ 27,315,423,531,639,…\frac { 2 } { 7 } , \frac { 3 } { 15 } , \frac { 4 } { 23 } , \frac { 5 } { 31 } , \frac { 6 } { 39 } , \ldots72​,153​,234​,315​,396​,… ​

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) a1=−8,ak+1=−3aka _ { 1 } = - 8 , a _ { k + 1 } = - 3 a _ { k }a1​=−8,ak+1​=−3ak​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Select the first five terms of the sequence defined recursively. $a _ { 1 } = 54 , a _ { k + 1 } = a _ { k } - 2$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Evaluate the sum. $\sum _ { k = 1 } ^ { 6 } 5 k$

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) $a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5$

Use a calculator to find the sum.Round to four decimal places. $\sum _ { k = 4 } ^ { 12 } \frac { - 1 } { k + 2 }$

Use sigma notation to write the sum. $\frac { 1 } { 3 \cdot 2 } + \frac { 1 } { 4 \cdot 3 } + \ldots + \frac { 1 } { 8 \cdot 7 }$

Select the first five terms of the sequence.(Assume that n begins with 1. ) $a _ { n } = \frac { 8 } { 5 }$

Assume that the sequence is defined recursively.Find the first three terms of the sequence. $a _ { 1 } = 3 \text { and } a _ { n + 1 } = 4 a _ { n } ^ { 2 }$

Select the first five terms of the sequence defined recursively. $a _ { 1 } = 23 , a _ { k + 1 } = a _ { k } + 3$

Find the sum. $\sum _ { i = 1 } ^ { 5 } - 3 i + 4$

Select the first five terms of the sequence.(Assume that n begins with 1. ) $a _ { n } = ( - 1 ) ^ { n } \left( \frac { n } { n + 11 } \right)$

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. ) $1,7,13,19,25 , \ldots$

Select the first five terms of the sequence.(Assume that n begins with 1. ) $a _ { n } = \left( \frac { 1 } { 5 } \right) ^ { n }$

Simplify the factorial expression. $\frac { 9 ! } { 6 ! }$

Use sigma notation to write the sum. $\frac { 1 } { 3 ( 1 ) } + \frac { 1 } { 3 ( 2 ) } + \frac { 1 } { 3 ( 3 ) } + \ldots + \frac { 1 } { 3 ( 9 ) }$

Find the indicated term of the sequence. $a _ { n } = ( - 1 ) ^ { n } ( 4 n - 2 )$ $a _ { 27 } =$ ?

Write an expression for the apparent nth term of the sequence.(Assume that n begins with 1. ) $\frac { 2 } { 7 } , \frac { 3 } { 15 } , \frac { 4 } { 23 } , \frac { 5 } { 31 } , \frac { 6 } { 39 } , \ldots$

Write the first five terms of the sequence defined recursively.Use the pattern to write the nth term of the sequence as a function of n.(Assume that n begins with 1. ) $a _ { 1 } = - 8 , a _ { k + 1 } = - 3 a _ { k }$