Question

Assertion (A): The numbers $$a, b, c$$ are in AP if and only if $$2b = (a + c).$$

Reason (R): The sum of the first $$n$$ odd natural numbers is $$n^{2}.$$

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 but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. For three numbers to be in AP, their consecutive differences must be equal: $$b-a = c-b,$$ which rearranges to $$2b = a+c.$$

Step 2 – Reason: R is true. The sum of the first $$n$$ odd natural numbers is $$n^{2}.$$

Step 3 – Link: The reason concerns the sum of odd natural numbers and does not explain the condition for three numbers to be in AP.