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 n^th term, and it will validate the statement for each and every term.

N^th term is given by ar^n-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.