등차수열 분수 합

[출제의도] 등차수열의 성질 이해하기 등차수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공차를 $d$라 하면 $a + 8 d = 2 \left( a + 2 d \right)$, $a = 4 d$ $\displaystyle \sum\limits _{n = 1} ^{24} \frac{\left( a _{n + 1} - a _{n} \right) ^{2}}{a _{n} a _{n + 1}}$ $\displaystyle = \sum\limits _{n = 1} ^{24} \frac{d ^{2}}{a _{n} a _{n + 1}}$$\displaystyle = d ^{2} \sum\limits _{n = 1} ^{24} \frac{1}{d} \left( \frac{1}{a _{n}} - \frac{1}{a _{n + 1}} \right)$ $\displaystyle = d \left\{ \left( \frac{1}{a _{1}} - \frac{1}{a _{2}} \right) + \left( \frac{1}{a _{2}} - \frac{1}{a _{3}} \right) + \cdots + \left( \frac{1}{a _{24}} - \frac{1}{a _{25}} \right) \right\}$ $\displaystyle = d \left( \frac{1}{a _{1}} - \frac{1}{a _{25}} \right)$$\displaystyle = d \left( \frac{a _{25} - a _{1}}{a _{1} \times a _{25}} \right)$ $\displaystyle = d \times \frac{24 d}{a \times \left( a + 24 d \right)} = \frac{24 d ^{2}}{4 d \times 28 d} = \frac{3}{14}$