Hit The Ceiling! Hit The Floor!

Algebra Level 3

$\large \lceil{x}\rceil\lfloor{x}\rfloor = x^2$

If $x$ is any positive irrational number that satisfies the equation above, then is it true that $\dfrac{x^2}{2}$ will be a triangular number?

False True

1 solution

Worranat Pakornrat
Mar 11, 2016

In the ceiling & floor function, for an irrational number $x$ , $\left\lceil{x}\right\rceil - \left\lfloor{x}\right\rfloor = 1$ .

Hence, if $\left\lfloor{x}\right\rfloor = n$ for any integer n, then $\left\lceil{x}\right\rceil = n + 1$ .

As a result, $x^2 = n(n + 1)$ ; $\dfrac{x^2}{2} = \dfrac{n(n+1)}{2}$ , which is a formula for a triangular number, which equals to $1 + 2 + 3 + \cdots + n$ for some integer n.

Therefore, if $x$ is an irrational number that satisfies the equation, then $\dfrac{x^2}{2}$ will be a triangular number.

Great question!

Anik Mandal - 5 years, 3 months ago

Thanks! ;)

Worranat Pakornrat - 5 years, 3 months ago

Hit The Ceiling! Hit The Floor!

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