Find the sum to n terms of the A.P., whose kth term is 5k + 1.
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 $d=5$

$a-d=1$

$\Rightarrow a-5=1$

$\Rightarrow a=6$

$S_{n}=\frac{n}{2}[2 a+(n-1) d]$

$=\frac{n}{2}[2(6)+(n-1)(5)]$

$=\frac{n}{2}[12+5 n-5]$

$=\frac{n}{2}(5 n+7)$

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