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in Arithmetic Progression by (44.5k points)
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In an A.P. Sn denotes the sum to first n terms, if Sn = n2p and Sm = m2p (m  n) prove that Sp = p3.

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Given: Sn = n2p and Sm = m2p

To Prove: Sp = p3

We know that,

⇒ 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

⇒ Sp = p3

Hence Proved

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