To find the $n$-th term of the sequence, let’s analyze the sum formula provided:

Given that the sum of the first $n$ terms, $S_n$, is:

$S_n = \frac{5n + 3n^2}{2}$

To find the $n$-th term of the sequence, $a_n$, we can use the formula:

$a_n = S_n - S_{n-1}$

Step 1: Calculate $S_{n-1}$

To find $S_{n-1}$, substitute $n-1$ in place of $n$ in $S_n$:

$S_{n-1} = \frac{5(n-1) + 3(n-1)^2}{2}$

Expanding this:

$S_{n-1} = \frac{5n - 5 + 3(n^2 - 2n + 1)}{2}$ $= \frac{5n - 5 + 3n^2 - 6n + 3}{2}$ $= \frac{3n^2 - n - 2}{2}$

Step 2: Find $a_n = S_n - S_{n-1}$

Now we can calculate $a_n$ by subtracting $S_{n-1}$ from $S_n$:

$a_n = \frac{5n + 3n^2}{2} - \frac{3n^2 - n - 2}{2}$

Combining terms:

$a_n = \frac{5n + 3n^2 - 3n^2 + n + 2}{2}$ $= \frac{6n + 2}{2}$ $= 3n + 1$

Answer

The $n$-th term, $a_n$, of the sequence is:

$a_n = 3n + 1$

Would you like further clarification on this solution?

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Here are five related questions that build on this topic:

1. How do you find the $n$-th term if the sum of the first $n$ terms is in a different form?
2. How would you approach finding the $n$-th term if the series were defined by a recursive formula?
3. What if the sum formula had a higher power of $n$ — how would that change the approach?
4. How does this approach apply to an arithmetic or geometric series?
5. How would we derive the sum formula if only the $n$-th term formula was given?

Tip: When finding the $n$-th term from a sum formula, always look at the difference $S_n - S_{n-1}$ to extract each term accurately.