$${1^3} - {2^3} + {3^3} - {4^3} + ..... + {9^3} = $$

Solution :
$$\eqalign{ & {1^3} - {2^3} + {3^3} - {4^3} + ..... + {9^3} \cr & = \,\,{1^3} + {2^3} + {3^3} + ..... + {9^3} - 2\left( {{2^3} + {4^3} + {6^3} + {8^3}} \right) \cr & = {\left[ {\frac{{9 \times 10}}{2}} \right]^2} - {2.2^3}\left[ {{1^3} + {2^3} + {3^3} + {4^3}} \right] \cr & = {\left( {45} \right)^2} - 16.{\left[ {\frac{{4 \times 5}}{2}} \right]^2} \cr & = 2025 - 1600 \cr & = 425 \cr} $$