In an AP, If `S_(n)=3n^(2)+5n` and `a_(k)=164,` then find the value of k.

Explanation

To solve the problem, we need to find the value of \( k \) such that the \( k \)-th term of the arithmetic progression (AP) is equal to 164, given that the sum of the first \( n \) terms of the AP is \( S_n = 3n^2 + 5n \).

### Step-by-Step Solution:

1. **Understanding the Formula for the \( n \)-th Term**:
The \( n \)-th term \( a_n \) of an AP can be expressed in terms of the sum of the first \( n \) terms \( S_n \) and the sum of the first \( n-1 \) terms \( S_{n-1} \):
\[
a_n = S_n - S_{n-1}
...