# If the pth, qth and rthterms of a G.P are a, b and c, respectively. Prove thataq – r × br – p × cP – q = 1.

Given pth, qth and rthterms of a G.P are a, b and c, respectively

Here

ap = a = arp-1

aq = b = arq-1

ar = c = arr-1

Now,

aq – r × br – p × cP – q = (arp - 1)q - r × (arq - 1)r - p × (arr-1)p – q

aq – r × br – p × cP – q = (a(q - r) × r(p – 1)(q-r)) × (a(r - p) × r(q – 1)(r - p)) × (a(p - q) × r(r – 1)(p - q))

aq – r × br – p × cP – q = (a(q – r) × r(pq – q - pr +r)) × (a(r - p) × r(qr – r - pq + p )) × (a(p - q) × r(pr – p – qr + q))

aq – r × br – p × cP – q = (a(q – r + r – p + p - q) × r(pq – q – pr +r + qr - r –pq + p +pr – p –qr + q))

aq – r × br – p × cP – q = (a0 × r0) = 1

aq – r × br – p × cP – q = 1

Hence proved.

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