Question

Assertion (A): The sum of first n natural numbers is given by $$S_{n} = \frac{1}{2}n(n + 1).$$

Reason (R): In an AP with first term a and common difference d, the sum of n terms is given by $$S_{n} = \frac{n}{2} \times [2a + (n - 1)d].$$

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. The natural numbers form an AP with first term $$a = 1$$ and common difference $$d = 1$$.

Step 2 – Reason: R is true. It gives the sum of $$n$$ terms of an AP.

Step 3 – Link: Applying the AP sum formula to the natural numbers gives $$S_{n} = \frac{n}{2}[2 + (n - 1)] = \frac{1}{2}n(n + 1),$$ so R correctly explains A.