Question

The sum $\sum_{n=1}^{10} \frac{528}{n(n+1)(n+2)}$ is equal to:

• A. 65
• B. 130
• C. 220
• D. 440

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to find the value of a finite sum. The term inside the summation has a denominator that is a product of three consecutive integers, which suggests that the sum can be simplified using the method of differences, also known as a telescoping series.
Step 2: Key Formula or Approach:
The core idea is to express the general term as a difference of two consecutive terms of a new sequence. For terms of the form $\frac{1}{n(n+1)...(n+k)}$, we can use a partial fraction-style decomposition. The key identity here is:
\[ \frac{1}{n(n+1)(n+2)} = \frac{1}{2} \left[ \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right] \] When summed, the intermediate terms will cancel out.
Step 3: Detailed Explanation:
Let the required sum be $S$. We can start by factoring out the constant:
\[ S = 528 \sum_{n=1}^{10} \frac{1}{n(n+1)(n+2)} \] Now, apply the identity to decompose the fraction:
\[ S = 528 \sum_{n=1}^{10} \frac{1}{2} \left[ \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right] \] \[ S = 264 \sum_{n=1}^{10} \left[ \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right] \] Let's define a sequence $T_n = \frac{1}{n(n+1)}$. The expression in the brackets is then $T_n - T_{n+1}$. The sum is a telescoping series:
\[ \sum_{n=1}^{10} (T_n - T_{n+1}) = (T_1 - T_2) + (T_2 - T_3) + \dots + (T_{10} - T_{11}) \] All terms except the first and the last cancel out, leaving:
\[ \sum_{n=1}^{10} (T_n - T_{n+1}) = T_1 - T_{11} \] Now we calculate $T_1$ and $T_{11}$:
\[ T_1 = \frac{1}{1(1+1)} = \frac{1}{2} \] \[ T_{11} = \frac{1}{11(11+1)} = \frac{1}{11 \times 12} = \frac{1}{132} \] The value of the summation part is:
\[ T_1 - T_{11} = \frac{1}{2} - \frac{1}{132} = \frac{66 - 1}{132} = \frac{65}{132} \] Finally, we multiply by the constant factor we set aside earlier:
\[ S = 264 \times \left( \frac{65}{132} \right) \] Since $264 = 2 \times 132$, the calculation simplifies to:
\[ S = 2 \times 65 = 130 \] Step 4: Final Answer:
The value of the sum is 130.