The sum of the first 9 terms of an AP is 81 and that of its first 20 terms is 400. Find the first term and the common difference of the AP.

Let a be the first term and d be the common difference.

Given: S9 = 81, S20 = 400


Now, consider S9 = 81


⇒ (9/2)[2a + (9 - 1)d] = 81


⇒ (9/2)[2a + 8d] = 81


⇒ [2a + 8d] = 18 ………(1)


Now, consider S20 = 400


⇒ (20/2)[2a + (20 - 1)d] = 400


⇒ 10 × [2a + 19d] = 400


⇒ [2a + 19d] = 40 ………..(2)


Now, on subtracting equation (2) from equation (1), we get,


11d = 22


⇒ d = 2


∴ from equation (1), we get


a = 1/2 (18 - 8d)


⇒ a = 9 - 4d


⇒ a = 9 - 8


⇒ a = 1


∴ a = 1, d = 2


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