등비수열 망원 합

[출제의도] 여러 가지 수열의 합 이해하기 등비수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공비를 $r$라 하면 $a _{10} = 8 a _{4}$에서 $ar ^{9} = 8 ar ^{3}$, $ar ^{3} ( r ^{6} - 8 ) = 0$ $ar \ne 0$이므로 $r ^{6} = 8$ 즉, $\displaystyle r = \sqrt{2}$ ($\because a _{n} > 0$) $a _{3} + a _{7} = 10$에서 $2 a + 8 a = 10$이므로 $a = 1$ $\therefore$ $\displaystyle a _{n} = ( \sqrt{2} ) ^{n-1}$ $\displaystyle \sum\limits _{n = 1} ^{10} \frac{a _{n + 1} - a _{n}}{\left( a _{n} + 1 \right) \left( a _{n + 1} + 1 \right)}$ $\displaystyle = \sum\limits _{n = 1} ^{10} \frac{a _{n + 1} - a _{n}}{( a _{n + 1} + 1 ) - ( a _{n} + 1 )} \left( \frac{1}{a _{n} + 1} - \frac{1}{a _{n + 1} + 1} \right)$ $\displaystyle = \sum\limits _{n = 1} ^{10} \left( \frac{1}{a _{n} + 1} - \frac{1}{a _{n + 1} + 1} \right)$ $\displaystyle = \frac{1}{a _{1} + 1} - \frac{1}{a _{11} + 1}$$\displaystyle = \frac{1}{1 + 1} - \frac{1}{32 + 1} = \frac{31}{66}$ 따라서 $\displaystyle \sum\limits _{n = 1} ^{10} \frac{a _{n + 1} - a _{n}}{\left( a _{n} + 1 \right) \left( a _{n + 1} + 1 \right)} = \frac{31}{66}$