MHB How Do Floor Functions Affect the Results of Square Roots in Sequence Problems?

  • Thread starter Thread starter anemone
  • Start date Start date
AI Thread Summary
The discussion focuses on a problem involving the computation of a ratio of products of floor functions applied to the fourth roots of integers from 1 to 2016. Participants share their approaches to solving the problem, emphasizing the impact of floor functions on the results of square roots in sequence problems. Several members successfully provide correct solutions, highlighting different methods and reasoning. The problem illustrates the significance of understanding how floor functions can alter numerical outcomes in mathematical sequences. Overall, the thread serves as a platform for collaborative problem-solving and mathematical exploration.
anemone
Gold Member
MHB
POTW Director
Messages
3,851
Reaction score
115
Here is this week's POTW:

-----

Compute $$\frac{\left\lfloor{\sqrt[4]{1}}\right\rfloor \cdot \left\lfloor{\sqrt[4]{3}}\right\rfloor \cdot\left\lfloor{\sqrt[4]{5}}\right\rfloor \cdots \left\lfloor{\sqrt[4]{2015}}\right\rfloor}{\left\lfloor{\sqrt[4]{2}}\right\rfloor \cdot \left\lfloor{\sqrt[4]{4}}\right\rfloor \cdot\left\lfloor{\sqrt[4]{6}}\right\rfloor \cdots \left\lfloor{\sqrt[4]{2016}}\right\rfloor}$$.

-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
Congratulations to the following members for their correct solution!(Cool)

1. Olinguito
2. castor28
3. Opalg
4. lfdahl
5. kaliprasad

Solution from castor28:
The expression can be written as:
$$
P=\prod_{n=0}^{1007}{\frac{\lfloor\sqrt[4]{2n+1}\rfloor}{\lfloor\sqrt[4]{2n+2}\rfloor}}
$$
Each fraction in the product is different from $1$ only when $2n+2=a^4$ for some integer $a$ (necessarily even). In that case, the fraction is equal to $\dfrac{a-1}{a}$.
As $6^4 < 2016 < 7^4$, this happens for $a=2,4,6$, and the expression is equal to:
$$
P = \frac12\times\frac34\times\frac56= \bf\frac{5}{16}
$$
 
Back
Top