the sum of the first n terms of an infinite sequence is (5n+3n^2)/2. find the nth term whats the answer



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.

The sum of the first n terms of an infinite sequence is (5n+3n^2)/2. Find the nth term.

Learn how to find the nth term of a sequence given the sum formula S_n = (5n + 3n^2)/2. This step-by-step solution covers methods for deriving individual terms from cumulative sums and provides insights into sequence and series concepts, suitable for high school students in Grades 10-12.

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