귀납적 수열 합

[출제의도] 귀납적으로 정의된 수열 추론하기 $\displaystyle a _{1} = \frac{1}{2}$, $\displaystyle a _{2} = \mathit{-} \frac{1}{a _{1} - 1} = 2$, $\displaystyle a _{3} = \mathit{-} \frac{1}{a _{2} - 1} \mathit{=} \mathit{-} 1$, $\displaystyle a _{4} = \mathit{-} \frac{1}{a _{3} - 1} \mathit{=} \frac{1}{2} \mathit{=} a _{1}$, $\displaystyle a _{5} = \mathit{-} \frac{1}{a _{4} - 1} \mathit{=} 2 \mathit{=} a _{2}$, $\cdots$ 수열 $\left\{ a _{n} \right\}$은 모든 자연수 $n$에 대하여 $a _{n + 3} = a _{n}$, $\displaystyle a _{3 n - 2} + a _{3 n - 1} + a _{3 n} = \frac{3}{2}$이므로 $\displaystyle S _{3 n} = \frac{3}{2} n$, $\displaystyle S _{3 n + 1} = S _{3 n} + \frac{1}{2} = \frac{3}{2} n + \frac{1}{2}$, $\displaystyle S _{3 n + 2} = S _{3 n} + \frac{1}{2} + 2 = \frac{3}{2} n + \frac{5}{2}$ $\displaystyle 11 = \frac{3}{2} \times 7 + \frac{1}{2} \mathit{=} S _{3 \times 7} + \frac{1}{2} \mathit{=} S _{3 \times 7 + 1} = S _{22}$ 따라서 $m = 22$