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 $m$.

• A. $23$
• B. $27$
• C. $24$
• D. $25$

Answer 1

Answer: (B)

Let $S_n$ denote the sum of $n$ terms and $a_n$ be the $n^{th}$ term of the given $A.P$.
Then, $S_n=3n^2+5n$
$\Rightarrow S_{n-1}=3(n-1{)}^2+5(n-1)=3n^2-n-2$
Now, $a_n=S_n-S_{n-1}$
$\Rightarrow a_n=(3n^2+5n)-(3n^2-n-2)$
$\Rightarrow a_n=6n+2$
Now, $a_m=164$
$\Rightarrow 6m+2=164$
$\Rightarrow 6m=162$
$\Rightarrow m=27$