If \(\lfloor x \rfloor = 4,\) find the sum of all possible values of \(\lfloor 4x \rfloor.\)
Another way of writing \(\lfloor x\rfloor = 4\) defines x values in the interval of \(4 \leq x < 5\)
If we multiply this interval by 4, we have that \(16\leq 4x < 20\).
In order to get the sum, we have to acknowledge that our 4x is still being floored, given \(\lfloor 4x \rfloor\). Therefore, only integer values within the interval can be a result of this. For example, \(x = 4.625\) satisfies that \(\lfloor x \rfloor = 4\). Then, \(\lfloor 4x \rfloor = \lfloor 4*4.625\rfloor = \lfloor 18.5 \rfloor = 18\)
The integers within \([16, 20)\) are 16, 17, 18, and 19. \(16+17+18+19= 70\).
If \(\lfloor x \rfloor = 4, \) the sum of all possible values of \(\lfloor 4x \rfloor\) is \(70\).