If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is

Here, we are given an A.P. whose $n^{\text {th }}$ term is given by the following expression, $a_{n}=2 n+1$. We need to find the sum of first $n$ terms.

So, here we can find the sum of the $n$ terms of the given A.P., using the formula, $S_{n}=\left(\frac{n}{2}\right)(a+l)$

Where, a = the first term

l = the last term

So, for the given A.P,

The first term (a) will be calculated using in the given equation for nth term of A.P.

$a=2(1)+1$

$=2+1$

$=3$

Now, the last term (l) or the nth term is given

$l=a_{n}=2 n+1$

So, on substituting the values in the formula for the sum of n terms of an A.P., we get,

$S_{n}=\left(\frac{n}{2}\right)[(3)+2 n+1]$

$=\left(\frac{n}{2}\right)[4+2 n]$

$=\left(\frac{n}{2}\right)(2)(2+n)$

$=n(2+n)$

Therefore, the sum of the $n$ terms of the given A.P. is $S_{n}=n(2+n)$. So the correct option is (b).