$Therangeoff\left(x\right)=[1+\sin x]+[2+\sin \frac{x}{2}]+[3+\sin \frac{x}{3}]+\mathrm{...}+[n+\sin \frac{x}{n}]\forall x\in [0,\pi ],where[.]$  where denotes the greatest integer function, is

$\begin{array}{l}f\left(x\right)=[1+\sin x]+[2+\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.]+[3+\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$3$}\right.]+---+[n+\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$n$}\right.]\\ forx\in (0,\pi ),\sin x,\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$3$}\right.,---\sin \raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\in (0,1)\\ ifx=0thenf\left(x\right)=1+2+3+---+n\\ =\frac{n(n+1)}{2}\\ ifx=\pi thenf\left(x\right)=1+[2+1]+3+4---+n\\ =(1+2+3+4+---+n)+1\\ =\frac{n(n+1)}{2}+1\\ =\frac{n^2+n+2}{2}\\ Range=\{\frac{n(n+1)}{2},\frac{n^2+n+2}{2}\}\end{array}$