The sum of n terms of an A.P is 3n2 + n, find the nth term.


To find the \(n\)th term of an arithmetic progression (A.P.) when the sum of the first \(n\) terms is given, we can use the following relation:

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

where \(a_n\) is the \(n\)th term, \(S_n\) is the sum of the first \(n\) terms, and \(S_{n-1}\) is the sum of the first \(n-1\) terms.

Given:

\[
S_n = 3n^2 + n
\]

### Step 1: Find \(S_{n-1}\)
Substitute \(n-1\) in place of \(n\) in the sum formula:

\[
S_{n-1} = 3(n-1)^2 + (n-1)
\]

Expanding the square:

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

### Step 2: Find the \(n\)th term (\(a_n\))

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

Substitute the values of \(S_n\) and \(S_{n-1}\):

\[
a_n = (3n^2 + n) - (3n^2 - 5n + 2)
\]

Simplify the expression:

\[
a_n = 3n^2 + n - 3n^2 + 5n - 2
\]
\[
a_n = 6n - 2
\]

### Final Answer:
The \(n\)th term of the A.P. is given by:

\[
a_n = 6n - 2
\]

Would you like more details on how these steps were carried out or explore related concepts?

Here are some related questions you might find interesting:

1. How do you find the common difference in an A.P. given the sum of \(n\) terms?
2. What is the formula for the sum of \(n\) terms in an arithmetic progression?
3. How does the sum of terms change if the A.P. starts from a different initial term?
4. Can we derive the first term of the A.P. using the given sum formula?
5. How does the nth term formula change for geometric progressions?

**Tip:** When dealing with sums of sequences, it's often useful to express them in general forms to identify patterns in the sequence or find specific terms directly.

Learn how to find the nth term of an arithmetic progression (A.P.) when the sum of the first n terms is given as 3n^2 + n. This detailed solution covers the formula for the nth term and step-by-step calculation process. Suitable for students in Grades 9-12.

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