If the sum of first p terms of an AP is (ap2 + bp), find its common

Let Sp denotes the sum of first p terms of an AP.

Sum of first p terms = Sp = ap2 + bp

Then pth term is given by: ap = Sp - Sp - 1

ap = (ap2 + bp) - [a(p - 1)2 + b(p - 1)]

= (ap2 + bp) - [a(p2 + 1 - 2p) + bp - b]

= ap2 + bp - ap2 - a + 2ap - bp + b

= b - a + 2ap

Now, common difference = d = ap - ap - 1

= b - a + 2ap - [b - a + 2a(p - 1)]

= b - a + 2ap - b + a - 2ap + 2a

= 2a

common difference = 2a

ALITER: Let Sp denotes the sum of first p terms of an AP.

Sum of first p terms = Sp = ap2 + bp

Put p = 1, we get S1 = a + b

Put p = 2, we get S2 = 4a + 2b

Now S1 = a1

a2 = S2 - S1

a2 = 3a + b

Now, d = a2 - a1

= 3a + b - (a + b)

= 2a

Common difference = 2a

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