A Fractional Periodicity

Algebra Level 3

$\large f(x) = \lfloor x\rfloor + \lfloor 2x\rfloor + \lfloor 3x\rfloor + \cdots + \lfloor nx\rfloor - \dfrac{n(n+1)}2 x$

Find the fundamental period of the function above.

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

None of these choices

n

1

\frac1n

1 solution

Young Wolf
Jun 7, 2016

$f(x)$ $=$ $[x]+[2x]+[3x]+....+[nx]$ $-$ $(x+2x+3x+.....+nx)$

$=$ $-$ $({\displaystyle \{x\}} +{\displaystyle \{2x\}} +{\displaystyle \{3x\}} +.....{\displaystyle \{nx\}} )$

Period of $f(x)$ is LCM $(1,\frac{1}{2}$ , $\frac{1}{3}$ ,.... $\frac{1}{n})$

$=$ $1$