Question

If pth, qth, and rth terms of a GP are in GP, show that p, q, r are in AP.

Answer 1

Given;

pth, qth and rth terms are in G.P.

To prove: p, q and r are in A.P.

Proof : Let Ap be the pth term, Aq be the qth term and Ar be the rth term.

Since, pth, qth and rth terms are in G.P., then geometeric mean equal;

Aq = √(Ap)(Ar);

Aq.Aq = (Ap)(Ar)

(ar^q)(ar^q) = (ar^p)(ar^r); {since, pth term = Ap = ar^p, similarly others}

ar^2q = ar^(p+r);

r^2q = r^(p+r);

2q = p + r;

q = (p+r)/2.

Since, arithematic mean of p, q, and r equal;

q = (p + r)/2;

Hence, p, q, and r are in A.P.

That's all.