Question

In an A.P., the sum of first n terms is `n/2 (3n + 5)`. Find the 25^th term of the A.P.

Answer 1

Given, S_n = `n/2 (3n + 5)`

∴ `S_(n - 1) = (n - 1)/2 [3(n - 1) + 5]`

or `S_(n - 1) = (n - 1)/2 (3n + 2)`

Since, a_n = S_n – S_n–1

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

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

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

= `(8n)/2 - n + 1`

= 4n – n + 1

= 3n + 1

Now, a₂₅ = 3(25) + 1

or, a₂₅ = 75 + 1 = 76

Thus, 25^th term of AP. is 76.