유리화 합

[출제의도] 여러 가지 수열의 합을 구할 수 있는가? $a _{n} = 4 + \left( n - 1 \right) \times 1 = n + 3$이므로 $\displaystyle \sum\limits _{k = 1} ^{12} \frac{1}{\sqrt{a _{k + 1}} + \sqrt{a _{k}}}$ $\displaystyle = \sum\limits _{k = 1} ^{12} \frac{\sqrt{a _{k + 1}} - \sqrt{a _{k}}}{\left( \sqrt{a _{k + 1}} + \sqrt{a _{k}} \right) \left( \sqrt{a _{k + 1}} - \sqrt{a _{k}} \right)}$ $\displaystyle = \sum\limits _{k = 1} ^{12} \frac{\sqrt{a _{k + 1}} - \sqrt{a _{k}}}{a _{k + 1} - a _{k}}$ $\displaystyle = \sum\limits _{k = 1} ^{12} \left( \sqrt{a _{k + 1}} - \sqrt{a _{k}} \right)$ $\displaystyle = \left( \sqrt{a _{2}} - \sqrt{a _{1}} \right) + \left( \sqrt{a _{3}} - \sqrt{a _{2}} \right) + \cdots + \left( \sqrt{a _{13}} - \sqrt{a _{12}} \right)$ $\displaystyle = \sqrt{a _{13}} - \sqrt{a _{1}}$ $\displaystyle = \sqrt{16} - \sqrt{4}$ $= 4 - 2 = 2$