Q45 of 176 Page 8

In an A.P. Sn denotes the sum to first n terms, if Sn = n2p and Sm = m2p (m n) prove that Sp = p3.

Given: Sn = n2p and Sm = m2p


To Prove: Sp = p3


We know that,




⇒ 2np = [2a + (n – 1)d]


⇒ 2np – (n – 1)d = 2a …(i)


and



⇒ 2mp = 2a + (m – 1)d


⇒ 2mp – (m – 1)d = 2a …(ii)


From eq. (i) and (ii), we get


⇒ 2np – (n – 1)d = 2mp – (m – 1)d


⇒ 2np – nd + d = 2mp – md + d


⇒ 2np – nd = 2mp – md


⇒ md – nd = 2mp – 2np


⇒ d(m – n) = 2p(m – n)


⇒ d = 2p …(iii)


Putting the value of d in eq. (i), we get


⇒ 2np – (n – 1)(2p) = 2a


⇒ 2pn – 2pn + 2p = 2a


⇒ 2p = 2a …(iv)


Now, we have to find the Sp


[from (iii) & (iv)]




⇒ Sp = p3


Hence Proved


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