Question

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

Answer 1

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

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
`T_20  = (5xx20 )-1= 99`
Hence, the 20^th term is 99.