To solve the problem, we need to find the number of terms \( n \) in an arithmetic progression (AP) where the sum of the first \( n \) terms is 252, the first term \( a \) is -16, and the last term \( l \) is 72. ### Step 1: Understanding the Sum of an Arithmetic Progression The formula for the sum of the first \( n \) terms of an arithmetic progression is given by: \[ S_n = \frac{n}{2} \times (a + l) \] where: - \( S_n \) is the sum of the first \( n \) terms, - \( n \) is the number of terms, - \( a \) is the first term, - \( l \) is the last term. ### Step 2: Plugging in the Known Values From the problem, we know: - \( S_n = 252 \) - \( a = -16 \) - \( l = 72 \) Substituting these values into the sum formula: \[ 252 = \frac{n}{2} \times (-16 + 72) \] ### Step 3: Simplifying the Equation First, calculate \( -16 + 72 \): \[ -16 + 72 = 56 \] Now, substitute this back into the equation: \[ 252 = \frac{n}{2} \times 56 \] ### Step 4: Solving for \( n \) To isolate \( n \), we can first multiply both sides by 2: \[ 504 = n \times 56 \] Next, divide both sides by 56: \[ n = \frac{504}{56} \] Calculating \( \frac{504}{56} \): \[ n = 9 \] ### Conclusion Thus, the number of terms \( n \) in the series is **9**. Therefore, the correct option is: **D. 9** ### Step 5: Explanation of Other Options - **Option A (6)**: If \( n = 6 \), then substituting back into the sum formula would yield a sum much less than 252, as \( \frac{6}{2} \times 56 = 168 \). - **Option B (7)**: If \( n = 7 \), then \( \frac{7}{2} \times 56 = 196 \), which is still less than 252. - **Option C (8)**: If \( n = 8 \), then \( \frac{8}{2} \times 56 = 224 \), which is also less than 252. ### Common Pitfalls 1. **Miscalculating the sum**: Ensure that you correctly apply the sum formula and simplify accurately. 2. **Forgetting to isolate \( n \)**: When solving for \( n \), remember to perform operations on both sides of the equation correctly. 3. **Confusing the first and last terms**: Always double-check that you are using the correct values for \( a \) and \( l \). ### Revision Summary - The sum of the first \( n \) terms of an AP is calculated using \( S_n = \frac{n}{2} \times (a + l) \). - Substitute known values into the formula to find \( n \). - Ensure calculations are accurate and isolate \( n \) correctly. - The correct answer for the number of terms in this series is **9** (Option D).