IF the sum of first p terms of an AP is $(a p^{2} + b p),$ find its common difference.

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Solution

Given that the sum of p terms of an AP is $(a p^{2} + b p)$ .

Now, Put p = 1,2,3,......

$S_{1} = a (1)^{2} + b (1) S_{1} = a + b = a_{1} S_{2} = a (2)^{2} + b (2) S_{2} = 4 a + 2 b N o w, S_{2} = a_{1} + a_{2} 4 a + 2 b = a + b + a_{2} 4 a + 2 b - a - b = a_{2} 3 a + b = a_{2} C o m m o n d i f f e r e n c e, d = a_{2} - a_{1} d = 3 a + b - (a + b) d = 3 a + b - a - b d = 2 a$

Therefore, the common difference of the given AP is 2a.

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