If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to

Let first term = a and Common difference = d


∴ According to the question, S n = n 2 p


Sn = n/2 (2a + (n–1) d) = n2p


2a + (n–1) d = 2np……………….(1)


And Sm = m2p


Sm= m/2 (2a + (m–1) d) = m2p


2mp = (2a + (m–1) d)……………….(2)


Subtracting 2 from 1


2a + (n–1) d – 2a – (m–1) d = 2 np – 2 mp


d (n–1 –m + 1) = 2p (n– m)


d = 2p


putting value of d in (1)


2a + (n–1) 2p = 2np


a + (n–1)p = np


a = p


now Sp = p/2 (2a + (p–1) d)


putting value of a = p and d = 2p


Sp = p/2 (2p + (p–1) 2p)


Sp = p3

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