Question

# what is the proof of the theorem:$\left[x\right]+\left[x+1/n\right]+\left[x+2/n\right]+\dots \dots .+\left[x+\left(n-2\right)/n\right]+\left[x+\left(n-2\right)/n\right]\left[x+\left(n-1\right)/n\right]=\left[nx\right]$

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Solution

## Proof of the theorem $\left[x\right]+\left[x+1/n\right]+\left[x+2/n\right]+\dots \dots .+\left[x+\left(n-2\right)/n\right]+\left[x+\left(n-2\right)/n\right]\left[x+\left(n-1\right)/n\right]=\left[nx\right]$Since, the greatest integer is defined as follow:$\left[m+\left(p/q\right)\right]=m$Where $m$is an integer value.$\left(p/q\right)$is the fractional value less than 1Now,$\begin{array}{rcl}& & \left[x\right]+\left[x+1/n\right]+\left[x+2/n\right]+\dots \dots .+\left[x+\left(n-2\right)/n\right]+\left[x+\left(n-2\right)/n\right]\left[x+\left(n-1\right)/n\right]=x+x+x....n\mathrm{times}\end{array}\phantom{\rule{0ex}{0ex}}\begin{array}{rcl}& ⇒& \left[x\right]+\left[x+1/n\right]+\left[x+2/n\right]+\dots \dots .+\left[x+\left(n-2\right)/n\right]+\left[x+\left(n-2\right)/n\right]\left[x+\left(n-1\right)/n\right]=n*x\end{array}$$\therefore \left[x\right]+\left[x+1/n\right]+\left[x+2/n\right]+\dots \dots .+\left[x+\left(n-2\right)/n\right]+\left[x+\left(n-2\right)/n\right]\left[x+\left(n-1\right)/n\right]=\left[nx\right]$ is provedHence, The theorem is proved.

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