The sum of the cubes would look like this:
(-15)^3 + (-14)^3+..+(-1)^3 + 0^3 + 1^2 + ..14^3 + 15^3 + 16^3 + 17^3(−15)3+(−14)3+..+(−1)3+03+12+..143+153+163+173
(color(red)(Note:Note:Assuming -15 and 17 are included)
As the cube of a negative number is negative,
(-15)^3(−15)3 and 15^3153 would cancel each other out;
(-14)^3(−14)3 and 14^3143 would cancel each other out;
and so on...
cancel((-15)^3) + cancel((-14)^3)+..+cancel((-1)^3) + 0^3 + cancel(1^2) + ..cancel(14^3) + cancel(15^3) + 16^3 + 17^3
What would remain is :
0^3 + 16^3 + 17^3
= 16^3 + 17^3
WE know that color(blue)(a^3+b^3 = (a + b)(a^2 - ab + b^2)
= (16 + 17)(16^2 - (16*17) + 17^2)
= 33*(256 - 272 + 289)
= 33*273
color(green)( = 9009
(color(red)(Note:If -15 and 17 are not included, then the answer would be 15^3 + 16^3 = color(green)(7471)
