분수형 점화식과 수열의 합

$\displaystyle \frac{8}{a _{k}} = b _{k}$라 하면 주어진 관계식은 $\displaystyle \left( 4 - \frac{8}{b _{k + 1}} \right) \left( 2 + \frac{8}{b _{k}} \right) = 8$ 전개하여 정리하면 $\displaystyle b _{k + 1} = \frac{1}{2} b _{k} + 2$ 즉, $b _{1} - 4 = 4$, $\displaystyle ( b _{k + 1} - 4 ) = \frac{1}{2} ( b _{k} - 4 )$이므로 $\displaystyle b _{k} - 4 = 4 \left( \frac{1}{2} \right) ^{k - 1}$ , $\displaystyle b _{k} = 4 + 4 \left( \frac{1}{2} \right) ^{k - 1}$ $\displaystyle \sum\limits _{k = 1} ^{9} b _{k}$$\displaystyle = 36 + \frac{4 \left\{ 1 - \left( \frac{1}{2} \right) ^{9} \right\}}{1 - \frac{1}{2}}$ $\displaystyle = 36 + 8 \left\{ 1 - \left( \frac{1}{2} \right) ^{9} \right\} = 44 - \left( \frac{1}{2} \right) ^{9}$ 따라서 구하는 정수부분은 43