Question

Find the sum of $n$ terms of the $A.P.,$ whose $k^{th}$ term is $5k+1.$

Answer 1

Step $1:$ Finding $A.P$

$k^{th}$ term of $A.P.$ is $5k+1i.e.,a_k=5k+1$
Substitute $k=1,2,\mathrm{3......},$ we get
$a_1=5\cdot 1+1=6,a_2=5\cdot 2+1=11,a_3=5\cdot 3+1=16,.....$
$\therefore$ we have $A.P.,6,11,\mathrm{16...}$
Here, $a=6$ and $d=11-6=5$

Step $2:$ Finding sum of $n$ terms of $A.P$
As we know, $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$
$S_n=\frac{n}{2}[2\times 6+(n-1\left)5\right]$
$\Rightarrow S_n=\frac{n}{2}[12+5n-5]$
$\therefore S_n=\frac{n}{2}[5n+7]$

Final answer: Hence, Sum of $n$ terms of the $A.P.$ is $\frac{n}{2}[5n+7]$