If p, q, r are in AP, then prove that pth, qth and rth terms of any GP are in GP.

To prove: p^th, q^th and r^th terms of any GP are in GP.

Given: (i) p, q and r are in AP

The formula used: (i) General term of GP,

As p, q, r are in A.P.

⇒ q – p = r – q = d = common difference … (i)

Consider a G.P. with the first term as a and common difference R

Then, the p^th term will be

the q^th term will be

the r^th term will be

Considering p^th term and q^th term

From eqn. (i) q – p = d

Considering q^th term and r^th term

From eqn. (i) r – q = d

We can see that p^th, q^th and r^th terms have common ration i.e

Hence they are in G.P.

Hence Proved