oldheaven wrote:what is the minimum range of "n" corresponded to this inequality:
" [cubic root of 1]+[cubic root of 2]+...+[cubic root of n]>=1389 "
1)between 250 and 260
2)between 300 and 310
3)between 320 and 330
4)between 340 and 350
The value of the greatest integer function of each number will depend on which two successive perfect integer cubes the number falls between.
[cube root of n] for 1<=n<8 is 1
[cube root of n] for 8<=n<27 is 2
[cube root of n] for 27<=n<64 is 3
[cube root of n] for 64<=n<125 is 4
[cube root of n] for 125<=n<216 is 5
[cube root of n] for 216<=n<343 is 6
The sum of the series for n=215 is 1(7)+2(19)+3(37)+4(61)+5(91)=855. The sum of the numbers whose cube roots have a greatest integer of 6 is 6(127)=762. 762+855=1587, which is larger than 1389, so the value of n must be such that 216<=n<343. To find out how many 6's we need, solve the following inequality:
855+6x>=1389
6x>=534
x>=89
We need 89 such terms, plus the 215 that gave us 855, so n=89+215=304.
So the answer is
2
