가중합과 부분분수

[출제의도] 여러 가지 수열을 활용하여 문제해결하기 $a _{1} + 2 a _{2} + 3 a _{3} + \cdots + n a _{n} = 830 n$ $\cdots \cdots \text{㉠}$ $a _{1} + 2 a _{2} + 3 a _{3} + \cdots + \left( n - 1 \right) a _{n - 1} = 830 \left( n - 1 \right)$ $\cdots \cdots \text{㉡}$ $\text{㉠} - \text{㉡}$을 하면, $na _{n} = 830 n - 830 ( n - 1 )$ $= 830 ( n \geq 2 )$ $a _{1} = 830$ $\displaystyle \therefore a _{n} = \frac{830}{n}$ $( n \geq 1 )$ $\displaystyle \sum\limits _{k = 1} ^{829} \frac{a _{k}}{k + 1}$$\displaystyle \begin{aligned} & = \sum\limits _{k = 1} ^{829} \frac{830}{k ( k + 1 )} \end{aligned}$ $\displaystyle \begin{aligned} & = 830 \left\{ \left( 1 - \frac{1}{2} \right) + \left( \frac{1}{2} - \frac{1}{3} \right) + \cdots + \left( \frac{1}{829} - \frac{1}{830} \right) \right\} \\ & = 829 \end{aligned}$