1^3-2^3+3^3-4^3++9^3=

Question

$1^3-2^3+3^3-4^3+\cdots+9^3=$

Accepted Answer

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

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

$$\,\,\,\,\,\,\,\,\,\, - 2\left( {{2^3} + {4^3} + {6^3} + {8^3}} \right)$$

$$ = {\left[ {{{9 \times 10} \over 2}} \right]^2} - {2.2^3}\left[ {{1^3} + {2^3} + {3^3} + {4^3}} \right]$$

$$ = {\left( {45} \right)^2} - 16.{\left[ {{{4 \times 5} \over 2}} \right]^2}$$

$= 2025 - 1600 = 425$

Solutions · 1