(d) If \( \sum_{m=1}^{5}(x-3 m)=\sum_{m=1}^{8}(x-3 m) \), prove that \( \sum_{i m=1}^{13}(x-3 m)=0 \). (e) Determine the value of \( p \) if \( \sum_{k=1}^{\infty} 27 p^{k}=\sum_{t=1}^{12}(24-3 t) \).

Let's tackle each part of the problem step by step. --- ### **Problem (d)** **Given:** \[ \sum_{m=1}^{5}(x - 3m) = \sum_{m=1}^{8}(x - 3m) \] **Prove:** \[ \sum_{m=1}^{13}(x - 3m) = 0 \] **Solution:** 1. **Evaluate the Sums:** - **Left Side:** \[ \sum_{m=1}^{5}(x - 3m) = 5x - 3\sum_{m=1}^{5}m \] \[ \sum_{m=1}^{5}m = \frac{5 \times 6}{2} = 15 \] \[ \Rightarrow 5x - 3 \times 15 = 5x - 45 \] - **Right Side:** \[ \sum_{m=1}^{8}(x - 3m) = 8x - 3\sum_{m=1}^{8}m \] \[ \sum_{m=1}^{8}m = \frac{8 \times 9}{2} = 36 \] \[ \Rightarrow 8x - 3 \times 36 = 8x - 108 \] 2. **Set the Sums Equal and Solve for \( x \):** \[ 5x - 45 = 8x - 108 \] \[ 8x - 5x = -45 + 108 \] \[ 3x = 63 \quad \Rightarrow \quad x = 21 \] 3. **Compute the Sum from \( m=1 \) to \( 13 \):** \[ \sum_{m=1}^{13}(x - 3m) = 13x - 3\sum_{m=1}^{13}m \] \[ \sum_{m=1}^{13}m = \frac{13 \times 14}{2} = 91 \] \[ \Rightarrow 13 \times 21 - 3 \times 91 = 273 - 273 = 0 \] **Conclusion:** \[ \sum_{m=1}^{13}(x - 3m) = 0 \quad \text{is proven.} \] --- ### **Problem (e)** **Given:** \[ \sum_{k=1}^{\infty} 27p^{k} = \sum_{t=1}^{12}(24 - 3t) \] **Find:** \[ p \] **Solution:** 1. **Evaluate Both Sides:** - **Left Side:** This is an infinite geometric series. \[ \sum_{k=1}^{\infty} 27p^{k} = 27 \sum_{k=1}^{\infty} p^{k} \] For \( |p| < 1 \), the sum of the geometric series is: \[ \sum_{k=1}^{\infty} p^{k} = \frac{p}{1 - p} \] \[ \Rightarrow 27 \times \frac{p}{1 - p} = \frac{27p}{1 - p} \] - **Right Side:** This is a finite arithmetic series. \[ \sum_{t=1}^{12}(24 - 3t) = \sum_{t=1}^{12}24 - 3\sum_{t=1}^{12}t \] \[ \sum_{t=1}^{12}24 = 24 \times 12 = 288 \] \[ \sum_{t=1}^{12}t = \frac{12 \times 13}{2} = 78 \] \[ \Rightarrow 288 - 3 \times 78 = 288 - 234 = 54 \] 2. **Set the Two Expressions Equal and Solve for \( p \):** \[ \frac{27p}{1 - p} = 54 \] \[ 27p = 54(1 - p) \] \[ 27p = 54 - 54p \] \[ 27p + 54p = 54 \] \[ 81p = 54 \] \[ p = \frac{54}{81} = \frac{2}{3} \] **Conclusion:** \[ p = \frac{2}{3} \] --- **Final Answer:** Problem (e) Answer: The value of \( p \) is \( \dfrac{2}{3} \).