If `S_n = 5n^2 + 3n`, then find its `n^(th)` term.

Answer

To find the \( n^{th} \) term of the sequence defined by \( S_n = 5n^2 + 3n \), we can use the relationship between the \( n^{th} \) term \( a_n \) and the sums \( S_n \) and \( S_{n-1} \). The formula for the \( n^{th} \) term is given by:

\[ a_n = S_n - S_{n-1} \]

### Step-by-Step Solution:

1. **Calculate \( S_n \)**:
Given \( S_n = 5n^2 + 3n \).

2. **Calculate \( S_{n-1} \)**:
To find \( S_{n-1} \), substitute \( n-1 \) into the formula for \( S_n \):
\[ S_{n-1} = 5(n-1)^2 + 3(n-1) \]
Expanding this:
\[ S_{n-1} = 5(n^2 - 2n + 1) + 3(n - 1) \]
\[ = 5n^2 - 10n + 5 + 3n - 3 \]
\[ = 5n^2 - 7n + 2 \]

3. **Find \( a_n \)**:
Now, substitute \( S_n \) and \( S_{n-1} \) into the formula for \( a_n \):
\[ a_n = S_n - S_{n-1} \]
\[ = (5n^2 + 3n) - (5n^2 - 7n + 2) \]
Simplifying this:
\[ = 5n^2 + 3n - 5n^2 + 7n - 2 \]
\[ = (5n^2 - 5n^2) + (3n + 7n) - 2 \]
\[ = 10n - 2 \]

4. **Final Answer**:
Thus, the \( n^{th} \) term \( a_n \) is:
\[ a_n = 10n - 2 \]