Book Online Demo

Check Your IQ

Try Test

Courses

Offline Centres

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

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

answer is C.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

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.

Watch 3-min video & get full concept clarity

JEE

NEET

Foundation JEE

Foundation NEET

CBSE