Question

If the p^th, q^th and r^th terms of a GP are a, b, c respectively, show that a^q-r b^r-p c^p-q = 1.

Answer 1

t_p = AR^p-1 = a; t_q = AR^q-1 = b; t_r = AR^r-1 = c

a^q-r b^r-p c^p-q = (AR^p-1)^q-r (AR^q-1)^r-p (AR^r-1)^p-q

= A^q-rR^(p-1)(q-r)A^r-p R^(q-1)(r-p)A^p-q R^(r-1)(p-q)

= A^q-r+r-p+p-q R^{pq-q-rp+r+rq-r-pq+p+rp-p-rq+q}

= A⁰R⁰ = 1