Question

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

Answer 1

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

∴ `"s"_"n" = ((3"n"^2) /2 +(5"n")/2)`

⇒ ` "s"_("n"-1) = (3("n"-1)^2)/2 + (5 ("n"-1))/2`

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

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

∴ n^th term pf the AP, a_n 

= `"s"_"n" - "s"_("n"-1)`

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

= `(6"n"+2)/2`

= 3n + 2

Putting n = 25, we get

a₂₅ = 3 × 25 + 1 = 75 + 1 = 76

Hence, the n^th term is (3n + 2) and 25^th term is 76.