Question

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.

Answer 1

Given: Sum of first 9 terms S₉ = 81 and sum of first 20 terms S₂₀ = 400.

Step-wise calculation:

1. Use `S_n = n/2 [2a + (n - 1)d]`.

2. For n = 9:

`S_9 = 9/2 [2a + 8d] = 81` 

⇒ 9[2a + 8d] = 162

⇒ 2a + 8d = 18

⇒ a + 4d = 9   ...(1)

3. For n = 20:

`S_20 = 20/2 [2a + 19d] = 400`

⇒ 10[2a + 19d] = 400

⇒ 2a + 19d = 40   ...(2)

4. Subtract (1) × 2 from (2):

(2a + 19d) – (2a + 8d) = 40 – 18

⇒ 11d = 22

⇒ d = 2

5. Substitute d = 2 into (1):

a + 4(2) = 9

⇒ a + 8 = 9

⇒ a = 1

First term a = 1 and common difference d = 2.