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QUESTION

The sum to infinity of a geometric series is 32. The sum of the first four terms is 30 and all the terms are positive. What's the difference between the sum to infinity and the sum of the first eight terms?

The difference between the sum to infinity and the sum of the first eight terms is ##1/8##.

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

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

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

##32×(1-r^4)=30## or ##1-r^4=30/32=15/16## and ##r^4=1-15/16=1/16## or ##r=1/2##.

Hence, as ##a/(1-r)=32##, ##a=32×(1-1/2)=32×1/2=16## i.e. first term is ##16## 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##

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