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 provided data:
The arithmetic progression (AP) is given as $21, 18, 15, 12, \dots$.
- The first term ($a$) is 21.
- The common difference ($d$) is calculated as $18 - 21 = -3$.
The task is to evaluate the given assertion and its supporting reason.

Step 2: Verify the assertion (A):
We need to determine if the sum of the first 15 terms of this AP equals zero.
The formula for the sum of the first $n$ terms of an AP is:\[S_n = \frac{n}{2} \left[ 2a + (n - 1) d \right]\]With $a = 21$, $d = -3$, and $n = 15$, the sum is:\[S_{15} = \frac{15}{2} \left[ 2(21) + (15 - 1)(-3) \right]\]Performing the calculations:\[S_{15} = \frac{15}{2} \left[ 42 + 14(-3) \right] = \frac{15}{2} \left[ 42 - 42 \right] = \frac{15}{2} \times 0 = 0\]The sum of the first 15 terms is confirmed to be zero.
Hence, assertion (A) is true.

Step 3: Verify the reason (R):
The formula for the sum of the first $n$ terms of an AP, $S_n = \frac{n}{2} \left[ 2a + (n - 1) d \right]$, where $a$ is the first term and $d$ is the common difference, is correctly stated.
Therefore, reason (R) is also true.

Step 4: Final determination:
As both the assertion (A) and the reason (R) are true, the overall statement is validated.