Compute \(\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\)

Guest Dec 2, 2018

edited by
Guest
Dec 2, 2018

#3**+1 **

Note that

2^4 = 16

3^4 = 81

4^4 = 256

5^4 = 625

6^4 = 1296

7^4 = 2401

So each floor from (1)^.25 to ( 15)^.25 = 1

And floor (16)^.25 = 2

And each floor from (17)^.25 to (80)^.25 = 2

And floor (81)^.25 = 3

And each floor from (82)^.25 to (255)^.25 = 3

And floor (256)^.25 = 4

And each floor from (257)^.25 to (624)^.25 = 4

And floor (625)^.25 = 5

And each floor from (626)^.25 to (1295)^.25 = 5

And floor (1296)^.25 = 6

And each floor from (1297)^.25 to (2016)^.25 = 6

So...note the pattern.....

(1)^8 * (2)^32 * (3)^88 * (4)^184 * .... 1 * 3 * ......

______________________________ = ___________

(1)^7 * (2)^33 * (3)^87* (4)^185 * ....... 2 * 4 * .....

1 * 3 * 5 *

__________ =

2 * 4 * 6

15

____ =

48

5

__

16

CPhill Dec 2, 2018

#6**+2 **

I think your calculation is incorrect, the answer I got is \(\frac{1}{\lfloor \sqrt[4]{2016} \rfloor}\)

HelloWorld
Dec 3, 2018