If p, q, r are in AP, then prove that pth, qth and rth terms of any GP are in GP.
To prove: pth, qth and rth 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 pth term will be
the qth term will be
the rth term will be
Considering pth term and qth term
From eqn. (i) q – p = d
Considering qth term and rth term
From eqn. (i) r – q = d
We can see that pth, qth and rth terms have common ration i.e
Hence they are in G.P.
Hence Proved