Q. 234.1( 7 Votes )

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

Answer :

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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