Question

If the sum of n terms of an A.P is given by S_n = n² + n, then the common difference of the A.P is

• A. 4
• B. 1
• C. 2
• D. 6

Answer 1

The Correct Option is C

The sum of n terms of an A.P. is given by \(S_n = n^2 + n\).

We need to find the common difference, denoted by 'd'.

We know that the sum of the first n terms of an A.P. can also be represented as:

\(S_n = \frac{n}{2}[2a + (n-1)d]\)

where 'a' is the first term and 'd' is the common difference.

We are given \(S_n = n^2 + n = n(n+1)\).

Let's find \(S_1\) and \(S_2\):

\(S_1 = 1^2 + 1 = 2\)

\(S_2 = 2^2 + 2 = 6\)

Since \(S_1\) represents the sum of the first term, it is equal to the first term itself:

\(a = S_1 = 2\)

\(S_2\) represents the sum of the first two terms, so:

\(S_2 = a + (a+d) = 2a + d = 6\)

We know \(a = 2\), so:

\(2(2) + d = 6\)

\(4 + d = 6\)

\(d = 6 - 4\)

\(d = 2\)

Therefore, the common difference is 2.