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