Question:
Find the sum to $n$ terms of the A.P. whose $k^{k h}$ term is $5 k+1$.
Solution:
It is given that the $k^{\text {th }}$ term of the A.P. is $5 k+1$.
$k^{\text {th }}$ term $=a_{k}=a+(k-1) d$
$\therefore a+(k-1) d=5 k+1$
$a+k d-d=5 k+1$
Comparing the coefficient of $k$, we obtain $d=5$
$a-d=1$
$\Rightarrow a-5=1$
$\Rightarrow a=6$
$S_{n}=\frac{n}{2}[2 a+(n-1) d]$
$=\frac{n}{2}[2(6)+(n-1)(5)]$
$=\frac{n}{2}[12+5 n-5]$
$=\frac{n}{2}(5 n+7)$