_n=1^10 (528)/(n(n+1)(n+2)) is equal to:

Question

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

Accepted Answer

Using partial fractions, $\dfrac{1}{n(n+1)(n+2)} = \dfrac12\left(\dfrac{1}{n(n+1)}-\dfrac{1}{(n+1)(n+2)}\right)$, so
$$\frac{528}{n(n+1)(n+2)} = 264\left(\frac{1}{n(n+1)}-\frac{1}{(n+1)(n+2)}\right).$$

This telescopes:
$$S = 264\left(\frac{1}{1\cdot2}-\frac{1}{11\cdot12}\right) = 264\left(\frac12-\frac{1}{132}\right) = 264\cdot\frac{65}{132} = 2\times65 = 130.$$

Solutions · 1