Question:
If the sum of n terms of an AP is $\left(3 n^{2}+5 n\right)$ and its mth term is 164, find the value of m.
Solution:
To Find: m
Given: Sum of $n$ terms, $m^{\text {th }}$ term
Put n = 1 to get the first term
So $a_{1}=3+5=8$
Put n = 2 to get the sum of first and second term
So $a_{1}+a_{2}=12+10=22$
So $a_{2}=14$
Common difference = 14 - 8 = 6
$T_{n}=a+(n-1) d=8+(n-1) 6=6 n+2$
Now 6m + 2 = 164
Or m = 27
The value of m is 27.