lf a, b, c are pth, qth and rth terms of a GP, then (q$$ $$-$$ $$r) log a $$+$$ (r$$ $$-$$ $$p) log b $$+$$ (p$$ $$-$$ $$q) log c is equal to

Question

Infinity Learn · Accepted Answer

Let A and R be the first term and common ratio of the given GP. Then, a $$=$$ $$ARP-1$$⇒log⁡$$a=log$$⁡$$A+(p$$−$$1)log$$⁡$$R-----isimilarly$$, log⁡$$b=log$$⁡$$A+(q$$−$$1)log$$⁡$$R----iiand$$ log⁡$$c=log$$⁡$$A+(r$$−$$1)log$$⁡$$R-----iiinow$$ , (q$$ $$-$$ $$r) log a $$+$$ (r$$ $$-$$ $$p) log b $$+$$ (p$$ $$-$$ $$q) log $$c=(q$$−$$r){log$$⁡$$A+(p$$−$$1)log$$⁡$$R}+(r$$−$$p){log$$⁡$$A+(q$$−$$1)log$$⁡$$R}+(p$$−$$q){log$$⁡$$A+(r$$−$$1)log$$⁡$$R}$$  $$=log$$⁡A[q−$$r+r$$−$$p+p$$−q]$$+log$$⁡R[$$p(q$$−$$r)+q(r$$−$$p)+r(p$$−$$q)$$−(q$$−$$r)−(r$$−$$p)−(p$$−$$q)]  $$=log$$⁡A⋅$$0+log$$⁡R⋅$$0=0$$

lf a, b, c are pth, qth and rth terms of a GP, then (q - r) log a + (r - p) log b + (p - q) log c is equal to

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