시그마 등식

[출제의도] 합의 기호 $\displaystyle \sum\limits _{} ^{}$의 성질을 이해한다. $\displaystyle \sum\limits _{k = 1} ^{n + 1} k ^{2} - \sum\limits _{k = 1} ^{n} \left( k ^{2} + k \right)$ $\displaystyle = \sum\limits _{k = 1} ^{n} k ^{2} + ( n + 1 ) ^{2} - \sum\limits _{k = 1} ^{n} \left( k ^{2} + k \right)$ $\displaystyle = ( n + 1 ) ^{2} + \sum\limits _{k = 1} ^{n} k ^{2} - \sum\limits _{k = 1} ^{n} \left( k ^{2} + k \right)$ $\displaystyle = ( n + 1 ) ^{2} + \sum\limits _{k = 1} ^{n} \left\{ k ^{2} - \left( k ^{2} + k \right) \right\}$ $\displaystyle = ( n + 1 ) ^{2} - \sum\limits _{k = 1} ^{n} k$ $\displaystyle = ( n + 1 ) ^{2} - \frac{n ( n + 1 )}{2}$ $\displaystyle = \frac{( n + 1 ) ( n + 2 )}{2}$ 따라서 $\displaystyle \frac{( n + 1 ) ( n + 2 )}{2} = 78$, $n ^{2} + 3 n - 154 = 0$ $( n - 11 ) ( n + 14 ) = 0$ ∴ $n = 11$ (∵ $n$은 자연수)