시그마 계산

[출제의도] $\displaystyle \sum\limits _{} ^{}$의 뜻과 성질 이해하기 $\displaystyle \sum\limits _{k = 1} ^{n + 1} ( k + 2 ) ^{2} - \sum\limits _{k = 1} ^{n} ( k ^{2} + 4 )$ $\displaystyle = \sum\limits _{k = 1} ^{n} ( k + 2 ) ^{2} + \left\{ ( n + 1 ) + 2 \right\} ^{2} - \sum\limits _{k = 1} ^{n} ( k ^{2} + 4 )$ $\displaystyle = \sum\limits _{k = 1} ^{n} \left\{ ( k + 2 ) ^{2} - ( k ^{2} + 4 ) \right\} + ( n + 3 ) ^{2}$ $\displaystyle = \sum\limits _{k = 1} ^{n} 4 k + ( n + 3 ) ^{2}$ $\displaystyle = 4 \times \frac{n ( n + 1 )}{2} + ( n + 3 ) ^{2} = 389$ $3 n ^{2} + 8 n - 380 = 0$ $( 3 n + 38 ) ( n - 10 ) = 0$ 따라서 $n = 10$ ($\because$ $n$은 자연수)