# 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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