귀납적으로 정의된 수열

$a _{2} = a _{2} \times a _{1} + 1$ $a _{8} = a _{2} \times a _{4} + 1$ $= a _{2} \times ( a _{2} \times a _{2} + 1 ) + 1$ $= ( a _{2} ) ^{3} + a _{2} + 1$ $a _{15} = a _{2} a _{7} - 2$ $= a _{2} ( a _{2} \times a _{3} - 2 ) - 2$ $= a _{2} ( a _{2} ( a _{2} \times a _{1} - 2 ) - 2 ) - 2$ $= a _{2} ( a _{2} ( a _{2} - 3 ) - 2 ) - 2$ $= ( a _{2} ) ^{3} - 3 ( a _{2} ) ^{2} - 2 a _{2} - 2$ $a _{8} - a _{15} = 63$에서 $( a _{2} ) ^{2} + a _{2} - 20 = 0$ $a _{2} = - 5$또는 $4$ $0 < a _{1} < 1$을 만족하는 $a _{2} = 4$이고 $\displaystyle a _{1} = \frac{3}{4}$ $\displaystyle \frac{a _{8}}{a _{1}} = \frac{69}{\frac{3}{4}} = 92$