We need to prove that the product of the terms equidistant from the beginning and end is the product of first and last terms in a finite GP.

Let us first consider a finite GP.

A, AR, AR²….AR^{n -1}, ARⁿ.

Where n is finite.

Product of first and last terms in the given GP = A.ARⁿ

= A²Rⁿ → (a)

Now, n^th term of the GP from the beginning = AR^n-1 → (1)

Now, starting from the end,

First term = ARⁿ

Last term = A

So, an n^th term from the end of GP, = AR → (2)

So, the product of n^th terms from the beginning and end of the considered GP from (1) and (2) = (AR^n-1) (AR)

= A²Rⁿ → (b)

So, from (a) and (b) its proved that the product of the terms equidistant from the beginning and end is the product of first and last terms in a finite GP.