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: Understanding the given information:
We are given the arithmetic progression (AP): $21, 18, 15, 12, \dots$.
- The first term ($a$) is 21.
- The common difference ($d$) is $18 - 21 = -3$.
We need to verify the assertion and reason provided.

Step 2: Checking the assertion (A):
We need to find the sum of the first 15 terms of the AP and check if it equals zero.
The sum of the first $n$ terms of an AP is given by the formula: \[ S_n = \frac{n}{2} \left[ 2a + (n - 1) d \right] \] Substituting the values $a = 21$, $d = -3$, and $n = 15$: \[ S_{15} = \frac{15}{2} \left[ 2(21) + (15 - 1)(-3) \right] \] Simplifying: \[ S_{15} = \frac{15}{2} \left[ 42 + 14(-3) \right] = \frac{15}{2} \left[ 42 - 42 \right] = \frac{15}{2} \times 0 = 0 \] Thus, the sum of the first 15 terms is indeed zero.
Therefore, the assertion (A) is true.

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

Step 4: Conclusion:
Since both the assertion (A) and reason (R) are true, the statement is correct.