Question

The sum of the first n terms of an AP is `((3n^2)/2 + (5n)/2)`. Find its nth term and the 25th term.

Answer 1

Let S_n denotes the sum of first n terms of the AP.

∴ `S_n = ((3n^2)/2 + (5n)/2)`

⇒ `S_(n - 1) = (3(n - 1)^2)/2 + (5(n - 1))/2`

= `(3(n^2 - 2n + 1))/2 + (5(n - 1))/2`

= `(3n^2 - n - 2)/2`

∴ n^th term of the AP,

a_n = S_n – S_{n – 1}

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

= `(6n + 2)/2`

= 3n + 2

Putting n = 25, we get

a₂₅ = 3 × 25 + 1

= 75 + 1

= 76

Hence, the nth term is (3n + 1) and 25th term is 76.