시그마의 성질

두 수열 $\left\{ a _{n} \right\}$, $\left\{ b _{n} \right\}$에 대하여 $\displaystyle \sum\limits _{k = 1} ^{10} \left( 2 a _{k} + b _{k} + k \right)$$\displaystyle = 2 \sum\limits _{k = 1} ^{10} a _{k} + \sum\limits _{k = 1} ^{10} b _{k} + \frac{10 \times 11}{2} = 60$ $\therefore$ $\displaystyle 2 \sum\limits _{k = 1} ^{10} a _{k} + \sum\limits _{k = 1} ^{10} b _{k} = 5$ $\cdots$$\cdots$ ㉠ $\displaystyle \sum\limits _{k = 1} ^{10} \left( a _{k} - 2 b _{k} + 1 \right)$$\displaystyle = \sum\limits _{k = 1} ^{10} a _{k} - 2 \sum\limits _{k = 1} ^{10} b _{k} + 10 = 10$ $\therefore$ $\displaystyle \sum\limits _{k = 1} ^{10} a _{k} = 2 \sum\limits _{k = 1} ^{10} b _{k}$ $\cdots$$\cdots$ ㉡ ㉠, ㉡을 연립하면 $\displaystyle \sum\limits _{k = 1} ^{10} a _{k} = 2$, $\displaystyle \sum\limits _{k = 1} ^{10} b _{k} = 1$ $\therefore$ $\displaystyle \sum\limits _{k = 1} ^{10} \left( a _{k} + b _{k} \right) = 2 + 1 = 3$