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

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Solution

$S_{p} = A P^{2} + b p A P = A P^{2} + b p - (A P^{2} - 2 A P + a) + b p - b A P = A P^{2} + b p \cdot A p^{2} + 2 A P - a + b p - b A P = 2 A P + 2 b p - a - b A P = 2 (a + b) - a - b A P = a + b p u t P = 2, a_{2} = 4 (a + b) - a - b a_{2} = 3 a + 3 b p u t p = 3, a L 3 = 6 (a + b) - a - b = 5 a + 5 b ∴ c o m m o n d i f f e r e n c e = a_{2} - a_{1} = 3 a + 3 b - a - b = 2 a + 2 b$

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