Mathematics - Functions Question with Solution | TestHub

Greatest integer function
$f\left(x\right)=\left[x\right]=\left\{\begin{array}{c}0 ; 0\le x<1\\ -1 ; -1\le x <0\\ -2 ; -2\le x<-1\end{array}\right.$

Given series
$S=\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\left[-\frac{1}{3}-\frac{3}{100}\right]+\dots +\left[-\frac{1}{3}-\frac{99}{100}\right]$
General term$T_r=\left[-\frac{1}{3}-\frac{r}{100}\right]=\left\{\begin{array}{c}-1 0\le r\le 66\\ -2 r>66\end{array}\right.$
$\Rightarrow S=\sum _{r=0}^{66}\left(-1\right)+\sum _{r=67}^{99}\left(-2\right)=\left(-67\right)+\left(-2\right)\times 33$
$= -133$