시그마 성질

[출제의도] $\displaystyle \sum\limits _{} ^{}$의 성질을 이용할 수 있는가? $\displaystyle \sum\limits _{k = 1} ^{10} \left( a _{k} + 1 \right) ^{2} = 28$에서 $\displaystyle \sum\limits _{k = 1} ^{10} \left\{ ( a _{k} ) ^{2} + 2 a _{k} + 1 \right\} = 28$ $\displaystyle \sum\limits _{k = 1} ^{10} ( a _{k} ) ^{2} + 2 \sum\limits _{k = 1} ^{10} a _{k} + \sum\limits _{k = 1} ^{10} 1 = 28$ $\displaystyle \sum\limits _{k = 1} ^{10} ( a _{k} ) ^{2} + 2 \sum\limits _{k = 1} ^{10} a _{k} = 18$ $\cdots \cdots$ ㉠ 또, $\displaystyle \sum\limits _{k = 1} ^{10} a _{k} ( a _{k} + 1 ) = 16$에서 $\displaystyle \sum\limits _{k = 1} ^{10} \left\{ ( a _{k} ) ^{2} + a _{k} \right\} = 16$ $\displaystyle \sum\limits _{k = 1} ^{10} ( a _{k} ) ^{2} + \sum\limits _{k = 1} ^{10} a _{k} = 16$ 이 식의 양변에 $2$를 곱하면 $\displaystyle 2 \sum\limits _{k = 1} ^{10} ( a _{k} ) ^{2} + 2 \sum\limits _{k = 1} ^{10} a _{k} = 32$ $\cdots \cdots$ ㉡ ㉡에서 ㉠을 변끼리 빼면 $\displaystyle \sum\limits _{k = 1} ^{10} ( a _{k} ) ^{2} = 14$