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.

Question

Philoid · Accepted Answer

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