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

Options

A. `k=17`
B. `k=27`
C. `k=37`
D. None

Explanation

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Step-by-Step Solution:

1. **Understand the Given Information**: We are given the sum of the first n terms of an arithmetic progression (AP) as \( S_n = 3n^2 + 5n \) and the k-th term \( a_k = 164 \).

2. **Find the (n-1)th Sum**: To find the k-th term \( a_k \), we need to calculate \( S_{n-1} \). This is done by substituting \( n-1 \) into the sum formula:
\[ S_{n-1} = 3(n-1)^2 + 5(n-1) \]
Expanding this:
\[ S_{n-1} = 3(n^2 - 2n + 1) + 5(n - 1) = 3n^2 - 6n + 3 + 5n - 5 \]
Simplifying further:
\[ S_{n-1} = 3n^2 - 6n + 5n + 3 - 5 = 3n^2 - n - 2 \]

3. **Calculate the k-th Term**: The k-th term \( a_k \) can be expressed as:
\[ a_k = S_k - S_{k-1} \]
Substituting the expressions we have:
\[ a_k = (S_k) - (S_{k-1}) = (3k^2 + 5k) - (3k^2 - k - 2) \]
Simplifying this:
\[ a_k = 3k^2 + 5k - 3k^2 + k + 2 = 6k + 2 \]

4. **Set Up the Equation**: We know from the problem that \( a_k = 164 \). Therefore, we can set up the equation:
\[ 6k + 2 = 164 \]

5. **Solve for k**: Rearranging the equation to find k:
\[ 6k = 164 - 2 \]
\[ 6k = 162 \]
Dividing both sides by 6:
\[ k = \frac{162}{6} = 27 \]

6. **Final Answer**: Thus, the value of \( k \) is \( 27 \).