Question

If the sum of $n$ terms of an A.P. is ${3n}^2+5n$ and its $m^{th}$ term is $164$. Find the value of $m$.

Answer 1

Let $a$ and $d$ be the first term and the common difference of the A.P. respectively

$a_m=a+(m-1)d=164...\left(1\right)$

Sum of $n$ terms , $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Here $\frac{n}{2}[2a+nd-d]={3n}^2+5n$

$\Rightarrow n(a-\frac{d}{2})+n^2\frac{d}{2}={3n}^2+5n$

Comparing the coefficient of $n\text{\&}{n}^2$ on both sides, we obtain

$\frac{d}{2}=3\Rightarrow d=6\text{\&}a-\frac{d}{2}\Rightarrow a-3=5\Rightarrow a=8$

Therefore, from (1) we obtain $8+(m-1)6=164$

$\Rightarrow (m-1)6=164-8=156\Rightarrow m-1=26\Rightarrow m=27$

Thus, the value of $m$ is $27.$