곱 꼴 점화식

$\displaystyle \frac{a _{n + 1}}{a _{n}} = \frac{n ^{2} + 2 n}{( n + 1 ) ^{2}}$$\displaystyle = \frac{n + 2}{n + 1} \times \frac{n}{n + 1}$ $\displaystyle \frac{a _{2}}{a _{1}} = \frac{3}{2} \times \frac{1}{2}$, $\displaystyle \frac{a _{3}}{a _{2}} = \frac{4}{3} \times \frac{2}{3}$, $\cdots$ , $\displaystyle \frac{a _{n}}{a _{n - 1}} = \frac{n + 1}{n} \times \frac{n - 1}{n}$ $\displaystyle \frac{a _{2}}{a _{1}} \times \frac{a _{3}}{a _{2}} \times \cdots \times \frac{a _{n}}{a _{n - 1}} = \frac{1 \times 3}{2 \times 2} \times \frac{2 \times 4}{3 \times 3} \times \cdots \times \frac{\left( n - 1 \right) \left( n + 1 \right)}{n \times n}$ $\therefore$ $\displaystyle a _{n} = \frac{1}{2} \times \frac{n + 1}{n} = \frac{n + 1}{2 n}$ 따라서 $100 a _{10} = 55$