세제곱수 교대합

$1 ^{3} - 2 ^{3} + 3 ^{3} - 4 ^{3} + \cdots + 19 ^{3}$ $= \left( 1 ^{3} + 3 ^{3} + \cdots + 19 ^{3} \right) - \left( 2 ^{3} + 4 ^{3} + \cdots + 18 ^{3} \right)$ $\displaystyle = \sum\limits _{k = 1} ^{10} \left( 2 k - 1 \right) ^{3} - \sum\limits _{k = 1} ^{9} \left( 2 k \right) ^{3}$ $\displaystyle = \sum\limits _{k = 1} ^{10} \left( 8 k ^{3} - 12 k ^{2} + 6 k - 1 \right) - 8 \sum\limits _{k = 1} ^{9} k ^{3}$ $\displaystyle = 8 \times 10 ^{3} + \sum\limits _{k = 1} ^{10} \left( - 12 k ^{2} + 6 k - 1 \right)$ $\displaystyle = 8000 - 12 \times \frac{10 \times 11 \times 21}{6} + 6 \times \frac{10 \times 11}{2} - 10$ $= 8000 - 4620 + 330 - 10$ $= 3700$