$S=1 \cdot 2+3-4+\ldots \ldots \ldots n$ terms

for n to be even, let $n=2 \mathrm{m}$

$S=1 \cdot 2+3 \cdot 4+\ldots \ldots \quad 2 m$ terms

$S=(1-2)+(3-4)+(5-6)+\ldots \ldots \ldots m$ terms

$S=(-1)+(-1)+(-1)+\ldots \ldots \ldots m$ terms

$S=-m=\frac{-n}{2}$

for $n$ to be odd let it be $2 \mathrm{m}+1$ so,

$\mathrm{S}=1-2+3-4+\ldots \ldots \ldots(2 \mathrm{m}+1)$ tems

$S=(1-2)+(3-4)+(5-6)+\ldots \ldots \ldots[(2 m-1)-2 m](2 m+1)$

$-m+2 m+1$

$ = (m + 1)\quad \left\langle {\begin{array}{*{20}{c}}
{n = 2m + 1}\\
{m = \frac{{n – 1}}{2}}
\end{array}} \right\rangle $

$\frac{\mathrm{n}-1}{2}+1$

$=\frac{n+1}{2}$