If the sum of first n terms of an A.P. is given by $S_n = \frac{n}{2} (3n + 1)$, then the first term of the A.P. is

If the sum of first n terms of an A.P. is given by $S_n = \frac{n}{2} (3n + 1)$, then the first term of the A.P. is

Given the sum of the first $ \displaystyle n $ terms of an A.P.:

$ \displaystyle S_n = \frac{n}{2} (3n + 1) $

The first term $ \displaystyle a $ is given by:

$ \displaystyle a = S_1 $

Substituting $ \displaystyle n = 1 $ in the given sum formula:

$ \displaystyle S_1 = \frac{1}{2} (3(1) + 1) $

$ \displaystyle = \frac{1}{2} (3 + 1) $

$ \displaystyle = \frac{1}{2} \times 4 $

$ \displaystyle = 2 $

Thus, the first term of the A.P. is $ \displaystyle 2 $.