2015/039) If p, q, r are in A.P., show that pth, qth, rth terms of any G.P. are themselves in G.P

$p, q, r$ are in AP

so $2q = p + r$

let for the gp 1st term is a and common factor is t

the pth term = $T_p = at^{p-1}$
qth terrm = $T_q = at^{q-1}$
r th term = $T_r = at^{r-1}$

pth term * rth term = $T_p * T_r = a^2t^{p+r-2} = a^2t^{2q-2} = (at^{q-1})^2 = T_q^2$ so they are in GP