Question

In an arithmetic progression the sum of the first n term,denoted by `S_n` is given by `S_n=n^2+8n` Find the first term and the common difference ?

Answer 1

The sum of the first n term of an AP,denoted by `S_n` is given by
`S_n=n^2+8n`

So

n th term of the series will be

`t_n=S_n-S_(n-1)=n^2+8n-((n-1)^2+8(n-1))`

`=>t_n=2n-1+8=2n+7`

Inserting `n=1,2,3....` we get

`"For "n=1tot_1=2*1+7=9`

`"For "n=2tot_2=2*2+7=11`

`"For "n=3tot_3=2*3+7=13`

Hence first term `t_1=9`
and common difference `d=t_2-t_1=11-9=2`

Check

Sum of first n terms

`S_n=n/2(2*9+(n-1)*2)`

`=>S_n=n/2(18+2n-2)`

`=>S_n=n/2(16+2n)`

`=>S_n=n^2+8n`