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Question

If Sn=n2p and Sm=m2p, mn in an A.P., prove that Sn=p3.

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Solution

Given,

Sn=n2[2a+(n1)d]=n2p

2a+(n1)d=2np...........(1)

Sm=m2[2a+(m1)d]=m2p

2a+(m1)d=2mp...........(2)

(1)-(2) gives,

2np2mp=2a+(n1)d2a+(m1)d

(nm)d=2(nm)p

d=2p

From (1)

2a+(n1)(2p)=2np

a+(n1)p=np

a=p

Sp=p2[2a+(p1)d]

=p2[2(p)+(p1)2p]

=p[p+p2p]

=p3

Hence proved.

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