Question

what is the proof of the theorem:

$\left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=\left[nx\right]$

Answer 1

Proof of the theorem $\left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=\left[nx\right]$

Since, the greatest integer is defined as follow:

$[m+(p/q\left)\right]=m$

Where $m$is an integer value.

$(p/q)$is the fractional value less than 1

Now,

$$\begin{gathered} \begin{array}{rcl}& & \left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=x+x+x....n\mathrm{times}\end{array} \\ \begin{array}{rcl}& \Rightarrow & \left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=n*x\end{array} \end{gathered}$$

$\therefore \left[x\right]+[x+1/n]+[x+2/n]+\dots \dots .+[x+(n-2)/n]+[x+(n-2)/n][x+(n-1)/n]=\left[nx\right]$ is proved

Hence, The theorem is proved.