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.


Means:


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 =


Hence,



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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