Q. Find common difference and first term of an A.S if the sum of the first 6 terms and 9 terms are 183 & 369 respectively.

Soln: 

Here,

Sum of the first 6 terms (S6) = 183

Sum of the first 9 terms (S6) = 369

Common difference (d) = ?

First Term (a) = ?

We have,

Sum of n terms of an AP is given by

Sn = (n/2)[2a+(n-1)d]

Thus,

Sum of the first 6 terms (S6) = 183

6/2[2a+5d] = 183

2a + 5d = 61 ................eqn(i)

Sum of the first 9 terms (S9) = 369

2a + 8d = 82.................eqn(ii)

Subtracting eqn(i) from (ii), we get,

3d = 21

d = 7

put value of d in eqn(ii),

2a + 8*7 = 82

2a = 82 - 56

2a = 26

a = 26/2

a = 13

Hence, Common difference (d) = 7 and First term (a) = 13  Ans.