If `S_(n) = 5n^(2) + 3n,` then `2^(nd)` term is …………… .

Answer

To find the \( n \)-th term of the sequence given the sum of the first \( n \) terms \( S_n = 5n^2 + 3n \), we will follow these steps:

### Step 1: Understand the formula for the \( n \)-th term
The \( n \)-th term \( a_n \) of a sequence can be found using the formula:
\[ a_n = S_{n} - S_{n-1} \]
where \( S_n \) is the sum of the first \( n \) terms and \( S_{n-1} \) is the sum of the first \( n-1 \) terms.

### Step 2: Calculate \( S_{n-1} \)
To find \( S_{n-1} \), we replace \( n \) with \( n-1 \) in the expression for \( S_n \):
\[ S_{n-1} = 5(n-1)^2 + 3(n-1) \]
Expanding this:
\[ S_{n-1} = 5(n^2 - 2n + 1) + 3(n - 1) = 5n^2 - 10n + 5 + 3n - 3 = 5n^2 - 7n + 2 \]

### Step 3: Find \( a_n \)
Now, we can find \( a_n \):
\[ a_n = S_n - S_{n-1} = (5n^2 + 3n) - (5n^2 - 7n + 2) \]
Simplifying this:
\[ a_n = 5n^2 + 3n - 5n^2 + 7n - 2 = 10n - 2 \]

### Step 4: Find the second term \( a_2 \)
To find the second term, we substitute \( n = 2 \):
\[ a_2 = 10(2) - 2 = 20 - 2 = 18 \]

### Final Answer
Thus, the second term \( a_2 \) is \( 18 \).

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