Question

If the sum of n terms of an A.P. is S_n = 3n² + 5n. Write its common difference.

Answer 1

Here, we are given,

`S_n = 3n^2 + 5n`

Let us take the first term as a and the common difference as d.

Now, as we know,

`a_n = S_n - S_(n-1)`

So, we get,

`a_n = (3n^2 + 5n) - [3(n-1)^2 + 5 (n-1)]`

`=3n^2 + 5n - [3(n^2 + 1 - 2n) + 5n - 5]      [\text{Using}  (a - b)^2= a^2 + b^2 - ab]`

`=3n^2 + 5n - (3n^2 + 3 - 6n + 5n - 5)`

`=3n^2 + 5n - 3n^2 - 3 + 6n - 5n + 5`

= 6n + 2                             ..................(1) 

Also,

`a_n = a + (n-1)d`

= a + nd - d 

= nd + (a - d)                     ...............(2) 

On comparing the terms containing n in (1) and (2), we get,

dn = 6n 

d = 6

Therefore, the common difference is d = 6.