귀납적으로 정의된 수열

$n ^{2} a _{n+1} = ( n + 1 ) ^{2} a _{n} + 2 n + 1$의 양변을 $n ^{2} ( n + 1 ) ^{2}$으로 나누면 $\displaystyle \frac{1}{( n + 1 ) ^{2}} a _{n+1} = \frac{1}{n ^{2}} a _{n} + \frac{2 n + 1}{n ^{2} ( n + 1 ) ^{2}}$ 이제, $\displaystyle b _{n} = \frac{a _{n}}{n ^{2}}$이라 두면 $\displaystyle b _{n+1} = b _{n} + \frac{2 n + 1}{n ^{2} ( n + 1 ) ^{2}}$ 그러므로 $\displaystyle b _{n} = b _{1} + \sum\limits _{k = 1} ^{n - 1} \frac{2 k + 1}{k ^{2} ( k + 1 ) ^{2}}$ $\displaystyle = \sum\limits _{k = 1} ^{n - 1} \left\{ \frac{1}{k ^{2}} - \frac{1}{( k + 1 ) ^{2}} \right\}$$\displaystyle \left( \because b _{1} = \frac{a _{1}}{1 ^{2}} = 0 \right)$ $\displaystyle = 1 - \frac{1}{n ^{2}}$ $\displaystyle = \frac{n ^{2} - 1}{n ^{2}}$ $\therefore a _{n} = n ^{2} b _{n} = n ^{2} - 1$ 따라서 $a _{20} = 20 ^{2} - 1 = 399$