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.