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