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 = 6 n + 2 $
Now, $a_m = 164$
$\Rightarrow 6m+ 2 = 164$
$\Rightarrow 6m = 162$
$\Rightarrow m = 27$