Question

The sum of the first n terms of an AP is `((5n^2)/2 + (3n)/2)`. Find the nth term and the 20th term of this AP.

Answer 1

`S_n = ((5n^2)/2 + (3n)/2) = 1/2 (5n^2 + 3n)`   ...(i)

Replacing n by (n – 1) in (i), we get:

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

= `1/2 xx [5n^2 - 10n + 5 + 3n - 3]`

= `1/2 xx [5n^2 - 7n + 2]`

∴ `T_n = S_n - S_(n - 1) `

= `1/2 (5n^2 + 3n) - 1/2 xx [5n^2 - 7n + 2]`

= `1/2 (10n - 2)`

= 5n – 1   ...(ii)

Putting n = 20 in (ii), we get

T₂₀ = (5 × 20) – 1 

= 100 – 1

= 99

Hence, the 20^th term is 99.