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

(m – n)d = 2p(m – n)
Now m is not equal to n
So d = 2p
Substituting in 1st equation we get

Hence proved.