Question

If a , b , c are respectively the p^th, q^th and r^th terms of a G . P show that (q – r) log a + (r – p) log b + (p – q) log c = 0

Answer 1

Let A be first term and R be the common ratio of the G.P.

Given a = p^th term of the G.P

General term of a G. P with first term A and common ratio R is t_n = AR^{n – 1}

∴ a = t_p = AR^{P – 1}

log a = log AR^p-1 

= log A + log R^p-1 

= log A + (p – 1) log R

b = q^th term of the G.P

b = t_q = AR^q-1

log b = log AR^q-1 

= log A + log R^q-1 

= log A + (q – r)log R

c = r^th term of the G.P

c = t_r = AR^r-1

log c = log AR^r-1 

= log A + log R^r-1 

= log A + (r – 1) log R

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

= (q – r) [log A + (p – 1) log R] + (r – p) [ log A + (q – 1) log R] + (P – q) [log A + (r – 1) log R]

= (q – r) log A + (q – r) (p – 1) log R + (r – p) log A + (r – p) (q – 1) log R + (P – q) log A + (p – q) (r – 1) log R

= [q – r + r – p + p – q] log A + [(q – r) (p – 1) + (r – p) (q – 1) + (p – q)(r – 1)] log R

= 0 × log A + [pq – q – rp + r + rq – r – pq + p + pr – p – rq + q] log R

= 0 × log R

= 0

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