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Question

If Pn is the sum of a GP upto n terms. Show that (1r)dpndr=n.Pn1(n1)Pn.

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Solution

Pn= Sum of n terms in G.P.
Pn=a(rn1)r1, a is a first term,r is a common ratio.
dPndr=anrn1(r1)a(rn1)(r1)2dPndr=anrnanrn1arn+a(r1)2dPndr=arn(n1)anrn1+a(r1)2dPndr=(n1)arnan+a+ananrn1(r1)2dPndr=(n1)a(rn1)(n)(a)(rn11)(r1)2dPndr=(n1)[a(rn1)r1]n[a(rn1)r1](r1)(r1)dPndr=(n1)PnnPn1(1r)dPndr=nPn1(n1)Pn

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