Question

Assertion (A): The sum of the first fifteen terms of the AP $21, 18, 15, 12, \dots$ is zero.
Reason (R): The sum of the first $n$ terms of an AP with first term $a$ and common difference $d$ is given by: $S_n = \frac{n}{2} \left[ a + (n - 1) d \right].$

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Answer 1

The Correct Option is A

Step 1: Analyze the assertion (A):
We are given the arithmetic progression (AP) \( 21, 18, 15, 12, \dots \), and we need to check if the sum of the first fifteen terms is zero.
The first term \( a = 21 \) and the common difference \( d = 18 - 21 = -3 \).
We are asked to find the sum of the first 15 terms, \( S_{15} \), of this AP.

Step 2: Use the formula for the sum of the first \( n \) terms of an AP:
The sum of the first \( n \) terms of an AP is given by the formula:
\[ S_n = \frac{n}{2} \left[ 2a + (n - 1) d \right] \] Substitute \( n = 15 \), \( a = 21 \), and \( d = -3 \) into the formula:
\[ S_{15} = \frac{15}{2} \left[ 2(21) + (15 - 1)(-3) \right] \] \[ S_{15} = \frac{15}{2} \left[ 42 + 14(-3) \right] \] \[ S_{15} = \frac{15}{2} \left[ 42 - 42 \right] = \frac{15}{2} \times 0 = 0 \] Thus, the sum of the first 15 terms is indeed zero.

Step 3: Analyze the reason (R):
The reason (R) states that the sum of the first \( n \) terms of an AP with first term \( a \) and common difference \( d \) is given by:
\[ S_n = \frac{n}{2} \left[ a + (n - 1) d \right] \] This is the correct formula for the sum of the first \( n \) terms of an arithmetic progression.

Step 4: Conclusion:
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion. Therefore, the correct answer is:
\[ \boxed{\text{Both (A) and (R) are true, and (R) is the correct explanation of (A)}} \]