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Question

Show that in an infinite G.P. with common ratio r(|r|<1), each term bears a constant ratio to the sum of all terms that follow it.

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Solution

Let us take a G.P. with terms a1,a2,a3,a4,.... and common ratio r(|r|<1).

Also, let us take the sum of all the terms following each term to be S1,S2,S3,S4,....

Now, S1=a2(1r)=ar(1r),

S2=a3(1r)=ar2(1r)

S3=a4(1r)=ar3(1r),

a1S1=aar(1r)=(1r)r,

a2S2=arar2(1r)=(1r)r,

a3S3=ar2ar3(1r)

It is clearly seen that the ratio of each term to the sum of all the terms following it is constant.


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