시그마 성질

$\displaystyle \sum\limits _{k = 1} ^{10} \left( a _{k} + k ^{2} \right) = 800$에서 $\displaystyle \sum\limits _{k = 1} ^{10} \left( a _{k} + k ^{2} \right)$$\displaystyle = \sum\limits _{k = 1} ^{10} a _{k} + \frac{10 \times 11 \times 21}{6}$$\displaystyle = \sum\limits _{k = 1} ^{10} a _{k} + 385$ 이므로 $\displaystyle \sum\limits _{k = 1} ^{10} a _{k} + 385 = 800$, $\displaystyle \sum\limits _{k = 1} ^{10} a _{k} = 415$ $\displaystyle \sum\limits _{k = 6} ^{10} \left( a _{k} - k ^{2} \right) = 50$에서 $\displaystyle \sum\limits _{k = 6} ^{10} \left( a _{k} - k ^{2} \right)$$\displaystyle = \sum\limits _{k = 6} ^{10} a _{k} - \left( \sum\limits _{k = 1} ^{10} k ^{2} - \sum\limits _{k = 1} ^{5} k ^{2} \right)$ $\displaystyle = \sum\limits _{k = 6} ^{10} a _{k} - \left( 385 - \frac{5 \times 6 \times 11}{6} \right)$ $\displaystyle = \sum\limits _{k = 6} ^{10} a _{k} - 330$ 이므로 $\displaystyle \sum\limits _{k = 6} ^{10} a _{k} - 330 = 50$, $\displaystyle \sum\limits _{k = 6} ^{10} a _{k} = 380$ $\therefore$$\displaystyle \sum\limits _{k = 1} ^{5} a _{k}$$\displaystyle = \sum\limits _{k = 1} ^{10} a _{k} - \sum\limits _{k = 6} ^{10} a _{k}$ $= 415 - 380$$= 35$