Q27 of 102 Page 410

If Sm = m2p and Sn = n2p, where m ≠ n in an AP then prove that Sp = p3.

Let the first term of the AP be a and the common difference be d


Given: Sm = m2p and Sn = n2p


To prove: Sp = p3


According to the problem


⇒2a + (m - 1)d = 2mp
⇒2a + (n - 1)d = 2np
Subtracting the equations we get,


(m - n)d = 2p(m - n)


Now m is not equal to n


So d = 2p
Substituting in 1st equation we get


2a + (m - 1)(2p) = 2mp
⇒a = mp - mp + p = p


Hence proved.


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