Question

If a, b, c are the p^th, q^th and r^th terms of a GP, show that

(q – r) log a + (r – p) log b + (p – q) log c = 0.

Answer 1

As per the question, a, b and c are the p^th, q^th and r^th term of GP. 

Let us assume the required GP as A, AR, AR² , AR³… 

Now, the nth term in the GP, an = AR^n-1 

p^th term, ap = AR^p-1 = a → (1) 

q^th term, aq = AR^q-1 = b → (2) 

r^th term, ar = AR^r-1 = c → (3)

Taking logarithm on both sides of equation (i), (ii) and (iii).

Now, multiply equation (4) with log c,

Now, add equations (7), (8) and (9).

On solving the above equation, we will get,

(p – q) log c + (q – r) log a + (r – p) log b = 0

Hence proved.