If S_n = n²p and in an A.P., prove that S_p = p³

given an AP whose sum of n terms is n²p and same AP with m terms whose sum is m²p

To prove: S_p = p³

we know that the sum of AP is given by the formula:

substituting the values in the above equation, we get

…. (i)

Similarly, for series with m terms

…(ii)

Subtracting (ii) from (i) we get

d = 2p

substituting d in (i) we get

a = p

Now using the sum formula for AP consisting of p terms we get

Substituting the values in the above equation

S_P = p³