Question 1

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  ​

Accepted Answer

A)    23,26,29,32,3523,26,29,32,3523,26,29,32,35  ​
B)    26,17,29,11,826,17,29,11,826,17,29,11,8  ​
C)    20,26,14,32,820,26,14,32,820,26,14,32,8  ​
D)    20,17,14,11,820,17,14,11,820,17,14,11,8  ​
E)    23,17,29,11,3523,17,29,11,3523,17,29,11,35  ​

Question 2

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​=  ?

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Question 5

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

Accepted Answer

A)    −85,−85,−85,−85,−85- \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 } , - \frac { 8 } { 5 }−58​,−58​,−58​,−58​,−58​  ​
B)    −58,58,−58,58,−58- \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 } , \frac { 5 } { 8 } , - \frac { 5 } { 8 }−85​,85​,−85​,85​,−85​  ​
C)    85,85,85,85,85\frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 } , \frac { 8 } { 5 }58​,58​,58​,58​,58​  ​
D)  0,1,2,3,4
E)    58,58,58,58,58\frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 } , \frac { 5 } { 8 }85​,85​,85​,85​,85​  ​

Question 6

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  ​

Accepted Answer

A)    56,58,60,62,6456,58,60,62,6456,58,60,62,64  ​
B)    52,58,50,62,6452,58,50,62,6452,58,50,62,64 
C)    56,52,60,48,6456,52,60,48,6456,52,60,48,64 
D)    54,58,50,62,4654,58,50,62,4654,58,50,62,46 
E)    54,52,50,48,4654,52,50,48,4654,52,50,48,46

Question 7

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​

Accepted Answer

A)    ∑n=161(n+2) !\sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 2 )  ! }∑n=16​(n+2) !1​ 
B)    ∑n=051(n+1) (n+2) \sum _ { n = 0 } ^ { 5 } \frac { 1 } { ( n + 1 )  ( n + 2 )  }∑n=05​(n+1) (n+2) 1​  ​
C)    ∑n=061n(n+1) \sum _ { n = 0 } ^ { 6 } \frac { 1 } { n ( n + 1 )  }∑n=06​n(n+1) 1​  ​
D)    ∑n=141(n+1) (n+2) \sum _ { n = 1 } ^ { 4 } \frac { 1 } { ( n + 1 )  ( n + 2 )  }∑n=14​(n+1) (n+2) 1​  ​
E)    ∑n=161(n+1) (n+2) \sum _ { n = 1 } ^ { 6 } \frac { 1 } { ( n + 1 )  ( n + 2 )  }∑n=16​(n+1) (n+2) 1​  ​

Question 8

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​

Accepted Answer

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

Question 9

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

Question 10

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,…  ​

Accepted Answer

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

Question 11

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

Accepted Answer

A)    281\frac { 2 } { 81 }812​ 
B)    24281\frac { 242 } { 81 }81242​ 
C)    8081\frac { 80 } { 81 }8180​  ​
D)    2681\frac { 26 } { 81 }8126​  ​
E)    8281\frac { 82 } { 81 }8182​

Question 12

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​  ​

Accepted Answer

A)    ∑i=19−13i\sum _ { i = 1 } ^ { 9 } - \frac { 1 } { 3 i }∑i=19​−3i1​ 
B)    ∑i=01013i\sum _ { i = 0 } ^ { 10 } \frac { 1 } { 3 i }∑i=010​3i1​ 
C)    ∑i=193i\sum _ { i = 1 } ^ { 9 } 3 i∑i=19​3i 
D)    ∑i=1913i\sum _ { i = 1 } ^ { 9 } \frac { 1 } { 3 i }∑i=19​3i1​ 
E)    ∑i=1−913i\sum _ { i = 1 } ^ { - 9 } \frac { 1 } { 3 i }∑i=1−9​3i1​

Question 13

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

Accepted Answer

A)  ​5040
B)  ​504
C)  ​3024
D)  ​72
E)  ​  32\frac { 3 } { 2 }23​

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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​

Accepted Answer

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

Question 17

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  ​

Accepted Answer

A)    15,110,115,1125,125\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 125 } , \frac { 1 } { 25 }51​,101​,151​,1251​,251​ 
B)    15,110,1125,120,13125\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 125 } , \frac { 1 } { 20 } , \frac { 1 } { 3125 }51​,101​,1251​,201​,31251​ 
C)    15,125,1125,1625,13125\frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 } , \frac { 1 } { 3125 }51​,251​,1251​,6251​,31251​ 
D)    15,110,115,120,125\frac { 1 } { 5 } , \frac { 1 } { 10 } , \frac { 1 } { 15 } , \frac { 1 } { 20 } , \frac { 1 } { 25 }51​,101​,151​,201​,251​ 
E)    1,15,125,1125,16251 , \frac { 1 } { 5 } , \frac { 1 } { 25 } , \frac { 1 } { 125 } , \frac { 1 } { 625 }1,51​,251​,1251​,6251​

Question 18

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​)   ​

Accepted Answer

A)    112,213,314,415,516\frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , \frac { 5 } { 16 }121​,132​,143​,154​,165​  ​
B)    −112,−213,−314,−415,−516- \frac { 1 } { 12 } , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , - \frac { 4 } { 15 } , - \frac { 5 } { 16 }−121​,−132​,−143​,−154​,−165​  ​
C)    −112,213,314,415,−516- \frac { 1 } { 12 } , \frac { 2 } { 13 } , \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }−121​,132​,143​,154​,−165​  ​
D)    −12,−213,−314,415,−516- 12 , - \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }−12,−132​,−143​,154​,−165​  ​
E)    −112,213,−314,415,−516- \frac { 1 } { 12 } , \frac { 2 } { 13 } , - \frac { 3 } { 14 } , \frac { 4 } { 15 } , - \frac { 5 } { 16 }−121​,132​,−143​,154​,−165​  ​

Question 19

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​,…  ​

Accepted Answer

A)    an=n+18n−8a _ { n } = \frac { n + 1 } { 8 n - 8 }an​=8n−8n+1​ 
B)    an=n−18n+1a _ { n } = \frac { n - 1 } { 8 n + 1 }an​=8n+1n−1​ 
C)    an=n+18n+1a _ { n } = \frac { n + 1 } { 8 n + 1 }an​=8n+1n+1​ 
D)    an=n−18n−1a _ { n } = \frac { n - 1 } { 8 n - 1 }an​=8n−1n−1​ 
E)    an=n+18n−1a _ { n } = \frac { n + 1 } { 8 n - 1 }an​=8n−1n+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. )   a1=19,ak+1=ak−5a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5a1​=19,ak+1​=ak​−5

Accepted Answer

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

Exam 52: Sequences and Series

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​,… ​

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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​ ​

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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 ​

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

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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​) ​

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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

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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,… ​

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

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

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

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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 ​

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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 ​

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

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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​

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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​

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

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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

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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​

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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​ ​

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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​= ?

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

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

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

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

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

Correct Answer
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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. ) $a _ { 1 } = 19 , a _ { k + 1 } = a _ { k } - 5$

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

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

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

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

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

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

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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. ) $a _ { 1 } = - 8 , a _ { k + 1 } = - 3 a _ { k }$

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

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

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

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

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

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