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The sum of the first n terms in the infinite geometric sequence \(\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}\) is \(\frac{255}{512}\). Find n.

 Jul 16, 2020
 #1
avatar+21958 
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The formula for the sum of the first n terms of a geometric series is:  Sum  =  a(1 - rn) / (1 - r)

where         a = first term = ¼         r = common ratio = ½          n = number of terms

 

--->     255/512  =  (¼)·( 1 - (½)n ) / ( 1 - ½ )

--->     255/512  =  (¼)·( 1 - (½)n ) / ( ½ )

--->     255/512  =  (½)·( 1 - (½)n )

--->     255/256  =  1 - (½)n 

--->           (½)n  =  1 - 255/256

--->           (½)n  =  1/256

--->           (½)n  =   (1/2)8

--->                n  =  8 

 Jul 16, 2020

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