조건에 따른 귀납적 수열

$a _{1} = 1$ $\displaystyle a _{2} = \sqrt[3]{2} \cdot 1 = 2 ^{\frac{1}{3}}$ $\displaystyle a _{3} = \sqrt[3]{2} \cdot 2 ^{\frac{1}{3}} = 2 ^{\frac{1}{3}} \cdot 2 ^{\frac{1}{3}} = 2 ^{\frac{2}{3}}$ $\displaystyle a _{4} = \sqrt[3]{2} \cdot 2 ^{\frac{2}{3}} = 2 ^{\frac{1}{3}} \cdot 2 ^{\frac{2}{3}} = 2$ $\displaystyle a _{5} = \frac{1}{2} \cdot 2 = 1$ $\displaystyle a _{6} = \sqrt[3]{2} \cdot 1 = 2 ^{\frac{1}{3}}$ $\cdots$ $a _{4n-3} = 1$, $\displaystyle a _{4n-2} = 2 ^{\frac{1}{3}}$ $\displaystyle a _{4n-1} = 2 ^{\frac{2}{3}}$, $a _{4n} = 2$ ($n = 1 , 2 , 3 \cdots$) ∴ $a _{112} = a _{4 \times 28} = 2$