수열 차의 제곱합

$\displaystyle \sum\limits _{k = 1} ^{n} \left( a _{k + 1} - a _{k} \right) ^{2} = 2 \left( 1 - \frac{1}{9 ^{n}} \right)$에서 $\displaystyle ( a _{n + 1} - a _{n} ) ^{2} = \sum\limits _{k = 1} ^{n} ( a _{k + 1} - a _{k} ) ^{2} - \sum\limits _{k = 1} ^{n - 1} ( a _{k + 1} - a _{k} ) ^{2}$ $\displaystyle = 2 \left( 1 - \frac{1}{9 ^{n}} \right) - 2 \left( 1 - \frac{1}{9 ^{n - 1}} \right)$ $\displaystyle = \frac{16}{9 ^{n}}$$( n \geq 2 )$ 이때, $n = 1$이면 $\displaystyle ( a _{2} - a _{1} ) ^{2} = 2 ( 1 - \frac{1}{9} ) = \frac{16}{9}$ 이므로 $\displaystyle ( a _{n + 1} - a _{n} ) ^{2} = \frac{16}{9 ^{n}}$ ($n \geq 1$) 한편, $a _{n + 1} > a _{n}$이므로 $\displaystyle a _{n + 1} - a _{n} = \frac{4}{3 ^{n}}$ $\therefore$ $\displaystyle a _{n + 1} = a _{n} + \frac{4}{3 ^{n}}$ 이때, $a _{1} = 10$이므로 $\displaystyle a _{n} = a _{1} + \sum\limits _{k = 1} ^{n - 1} \frac{4}{3 ^{k}}$ $\displaystyle = 10 + \frac{\frac{4}{3} \left\{ 1 - \left( \frac{1}{3} \right) ^{n - 1} \right\}}{1 - \frac{1}{3}}$ $\displaystyle = 10 + 2 \left\{ 1 - \left( \frac{1}{3} \right) ^{n - 1} \right\}$($n \geq 2$) 따라서, $\displaystyle \lim\limits _{n \rightarrow \infty} {a _{n}} = 12$