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: Examine Assertion (A): The provided arithmetic progression (AP) is \( 21, 18, 15, 12, \dots \). We must verify if the sum of its first fifteen terms equals zero. The initial term is \( a = 21 \), and the common difference is \( d = 18 - 21 = -3 \). The objective is to calculate the sum of the first 15 terms, denoted as \( S_{15} \).

Step 2: Apply the AP Sum Formula: The formula for the sum of the first \( n \) terms of an AP is \( S_n = \frac{n}{2} \left[ 2a + (n - 1) d \right] \). Substituting \( n = 15 \), \( a = 21 \), and \( d = -3 \) yields:
\[S_{15} = \frac{15}{2} \left[ 2(21) + (15 - 1)(-3) \right]\]\[S_{15} = \frac{15}{2} \left[ 42 + 14(-3) \right]\]\[S_{15} = \frac{15}{2} \left[ 42 - 42 \right] = \frac{15}{2} \times 0 = 0\]Consequently, the sum of the first 15 terms is confirmed to be zero.

Step 3: Examine Reason (R): 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 formula is indeed the correct one for calculating the sum of the first \( n \) terms of an arithmetic progression.

Step 4: Conclusion: Both the assertion (A) and the reason (R) are accurate, and reason (R) correctly substantiates assertion (A). The appropriate conclusion is:
\[\boxed{\text{Both (A) and (R) are true, and (R) is the correct explanation of (A)}}\]