A glossary of gauss functions

Algebra Level pending

If $\{a\}, \lfloor a \rfloor, a$ are of geometric progression , how many possible values are there for positive real number $a$ ?

Notations :

$\{\cdot\}$ denotes the fractional part function .
$\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 1.

1 solution

A Former Brilliant Member
Oct 11, 2019

Let $a=b+c$ where $b$ is the floor function of $a$ and $c$ is it's fractional part. Then $c(b+c)=b^2$ or $b=\dfrac{c(\sqrt 5+1)}{2}$ . Since $b$ is an integer and $0<c<1$ , therefore $c=\dfrac{\sqrt 5-1}{2},b=1$ and $a=\dfrac{\sqrt 5+1}{2}$

A glossary of gauss functions

The answer is 1.

1 solution

0 pending reports