If the sum of first p terms of an AP is ( $a p 2 + b p$ ), then what is the common difference?

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answer is C.

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Detailed Solution

Given that the sum of first p terms of an AP is

a p 2 + b p

.
Formula used for sum of first term of A.P,

S n = n 2 [2 a + (n − 1) d]

Consider,

S p

denotes the sum of first p terms of the AP.

S p = a p 2 + b p

S p − 1 = a (p − 1) 2 + b (p − 1) = a (p 2 − 2 p + 1) + b (p − 1) = a p 2 − (2 a − b) p + (a − b)

As the pth term of the AP,

a p = S p − S p − 1

= a p 2 + b p − a p 2 − (2 a − b) p + (a − b) = a p 2 + b p − a p 2 + (2 a − b) p − (a − b) = 2 a p − (a − b)

Let d is the common difference of the AP.

∴ d = a p − a p − 1 = [2 a p − (a − b)] − [2 a (p − 1) − (a − b)] = 2 a p − (a − b) − 2 a (p − 1) + (a − b) = 2 a

Therefore, the common difference of the AP is 2a.
Hence, option (3) is correct.

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