부분분수 합

[출제의도] 여러 가지 수열의 합 이해하기 수열 $\left\{ a _{n} \right\}$이 등차수열이므로 $a _{n} = 1 + \left( n - 1 \right) \times 3 = 3 n - 2$ $\displaystyle \sum\limits _{k = 1} ^{10} \frac{1}{a _{k} a _{k + 1}}$ $\displaystyle = \sum\limits _{k = 1} ^{10} \frac{1}{\left( 3 k - 2 \right) \left( 3 k + 1 \right)}$ $\displaystyle = \frac{1}{3} \sum\limits _{k = 1} ^{10} \left( \frac{1}{3 k - 2} - \frac{1}{3 k + 1} \right)$ $\displaystyle = \frac{1}{3} \left\{ \left( 1 - \frac{1}{4} \right) \mathit{+} \left( \frac{1}{4} - \frac{1}{7} \right) \mathit{\mathit{+}} \right.$$\displaystyle \left. \cdots \mathit{+} \left( \frac{1}{28} - \frac{1}{31} \right) \right\}$ $\displaystyle = \frac{1}{3} \left( 1 - \frac{1}{31} \right) = \frac{10}{31}$