Question

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

Answer 1

Given the sum of 9 terms of AP = 81
Recall that sum of n terms of AP is:

$S_n$ = $\frac{n}{2}$[ 2a+ (n-1)d]

⇒ 81 = $\frac{9}{2}$[ 2a+ (9-1)d]

⇒ 9 = $\frac{1}{2}$ [ 2a+8d]​​​​​​​​​​​​​​

⇒​​​​​​​ 18 = 2a + 8d ---------(i)

It is also given that sum of first 20 terms is 400

⇒​​​​​​​ 400 = $\frac{20}{2}$[ 2a+ (20-1)d]​​​​​​​

⇒​​​​​​​ ​​​​​​​400 = ​​​​​​​ 10 [ 2a + 19d]

⇒​​​​​​​ 40 = 2a + 19d ---------(ii)

Subtract (i) from (ii) we get;

⇒ 11d = 22

⇒ d = 2

Substitute d = 2 in (i), we get;
⇒ 2a + 8(2) = 18
⇒ 2a = 18 − 16 = 2
∴ a = 1
The first term is 1 and the common difference is 2.