If a, b, c are positive and are the pth, qth, and rth terms, respectively, of a G.P., then Δ=logap1logbq1logcr1 is

Let the first term of G.P. be A and common ratio be R. Then,a=ARp−1⇒ log⁡a=log⁡A+(p−1)log⁡R, etc. ⇒log⁡ap1log⁡bq1log⁡cr1=log⁡A+(p−1)log⁡Rp1log⁡A+(q−1)log⁡Rq1log⁡A+(r−1)log⁡Rr1                            =(p−1)log⁡Rp1(q−1)log⁡Rq1(r−1)log⁡Rr1 C1→C1−(log⁡A)C3=log⁡R(p−1)p1(q−1)q1(r−1)r1=log⁡Rpp1qq1rr1=0 C1→C1+C3