Answer :

Let a be the first term of GP.

Given common ratio = r

we can write GP as : a ,ar ,ar2 ,ar3

We need to proof that: each term bears a constant ratio to the sum of all terms that follow it.


Proving for each and every individual term will be a tedious and foolish job.

So we will prove this for the nth term, and it will validate the statement for each and every term.

Nth term is given by arn-1.

To prove:

We know that sum of an infinite GP is given by:

Sum of infinite GP = ,where a is the first term and k is the common ratio.

∴ arn + arn+1 + … ∞ = arn(1 + r + r2 +…∞)

Sum =


As the ratio is independent of the value of each and every term

And hence we say that it bears a constant ratio. Proved.

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