Question

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

• A. 2
• B. \( \frac{3}{2} \)
• C. 4
• D. \( \frac{5}{2} \)

Answer 1

The Correct Option is A

Given:
The sum of the first \(n\) terms of an arithmetic progression (A.P.) is:
\[ S_n = \frac{n}{2}(3n + 1) \] where \(S_n\) denotes the sum of the first \(n\) terms.

Step 1: Understand what is asked
We need to find the first term \(a\) of the A.P.

Step 2: Recall relation between sum and first term
The first term \(a\) is the sum of the first term, i.e.,
\[ a = S_1 \] This means, if we put \(n = 1\) in the sum formula \(S_n\), the result will give us the first term \(a\).

Step 3: Substitute \(n = 1\) in the sum formula
\[ S_1 = \frac{1}{2} (3 \times 1 + 1) = \frac{1}{2} (3 + 1) = \frac{1}{2} \times 4 = 2 \]

Step 4: Interpret the result
The value \(S_1 = 2\) means the first term \(a = 2\).

Final Answer:
\[ \boxed{2} \]