곱 점화식 급수

[출제의도] 수열의 귀납적 정의를 이용하여 일반항을 추측한다. $\displaystyle a _{1} = \frac{1}{2 ^{3}}$ $a _{1} a _{2} = 2 ^{1}$에서 $\displaystyle a _{2} = \frac{2}{a _{1}} = 2 ^{4}$ $a _{2} a _{3} = 2 ^{2}$에서 $\displaystyle a _{3} = \frac{2 ^{2}}{a _{2}} = \frac{1}{2 ^{2}}$ $a _{3} a _{4} = 2 ^{3}$에서 $\displaystyle a _{4} = \frac{2 ^{3}}{a _{3}} = 2 ^{5}$ $a _{4} a _{5} = 2 ^{4}$에서 $\displaystyle a _{5} = \frac{2 ^{4}}{a _{4}} = \frac{1}{2}$ $\vdots$ $\displaystyle \left\{ a _{2 n - 1} \right\} : \frac{1}{2 ^{3}} , \frac{1}{2 ^{2}} , \frac{1}{2 ^{1}} , \cdots$이므로 $\displaystyle \left\{ \frac{1}{a _{2 n - 1}} \right\} : 2 ^{3} , 2 ^{2} , 2 ^{1} , \cdots$ ∴ $\displaystyle \sum\limits _{n = 1} ^{\infty} \frac{1}{a _{2 n - 1}} = \frac{8}{1 - \frac{1}{2}} = 16$ [다른 풀이] $a _{n} a _{n+1} = 2 ^{n}$에서 $a _{n+1} a _{n+2} = 2 ^{n+1}$ $\displaystyle \frac{a _{n+1} a _{n+2}}{a _{n} a _{n+1}} =$$\displaystyle \frac{2 ^{n+1}}{2 ^{n}}$, $\displaystyle \frac{a _{n + 2}}{a _{n}} = 2$ $a _{n+2} = 2 a _{n}$ $\displaystyle a _{1} = \frac{1}{8}$이므로 $a _{2n-1} = a _{1} \cdot 2 ^{n-1}$$= 2 ^{n-4}$ ∴ $\displaystyle \sum\limits _{n = 1} ^{\infty} \frac{1}{a _{2 n - 1}}$$\displaystyle = \sum\limits _{n = 1} ^{\infty} \frac{1}{2 ^{n-4}} = \frac{8}{1 - \frac{1}{2}} = 16$