If sum of first four terms of a geometric series is 3030 and sum of the series to infinity is 3232, find the difference between the sum of the series to infinity and the sum of its first eight terms?

1 Answer
Sep 7, 2016

The difference between the sum to infinity and the sum of the first eight terms is 1/818.

Explanation:

A geometric series is one in which ratio of a term to its preceding term, generally described as rr, is always constant.

It is mentioned that sum to infinity is 3232, which means series is converging and r<1r<1. In a geometric series, if aa is the first term, such "sum to infinity" is given by a/(1-r)a1r. Hence, a/(1-r)=32a1r=32.

Further, sum of first nn terms is given by a×(1-r^n)/(1-r)a×1rn1r. As first four terms add up to 3030, a×(1-r^4)/(1-r)=30a×1r41r=30, but as a/(1-r)=32a1r=32, we have

32×(1-r^4)=3032×(1r4)=30 or 1-r^4=30/32=15/161r4=3032=1516 and r^4=1-15/16=1/16r4=11516=116 or r=1/2r=12.

Hence, as a/(1-r)=32a1r=32, a=32×(1-1/2)=32×1/2=16a=32×(112)=32×12=16 i.e. first term is 1616 and series is {16,8,4,2,...}.

First 8 terms add upto 16×(1-1/2^8)/(1-1/2) or

16(1-1/256)/(1/2)

= 16×2×(1-1/256)

= 32-1/8=255/8 or 31 7/8

Hence, the difference between the sum to infinity, which is 32 and the sum of the first eight terms is 1/8.