The value of $\left[\frac{{\displaystyle 1}}{{\displaystyle 2}}\right]+\left[\frac{{\displaystyle 1}}{{\displaystyle 2}}+\frac{{\displaystyle 1}}{{\displaystyle 100}}\right]+\left[\frac{{\displaystyle 1}}{{\displaystyle 2}}+\frac{{\displaystyle 2}}{{\displaystyle 100}}\right]+\cdots +\left[\frac{{\displaystyle 1}}{{\displaystyle 2}}+\frac{{\displaystyle 99}}{{\displaystyle 100}}\right]$ is

We have,

$\frac{1}{2}+\frac{\mathrm{n}}{100}<1 \text{if }\mathrm{n}\le 49$

$\therefore \left[\frac{{\displaystyle 1}}{{\displaystyle 2}}+\frac{{\displaystyle \mathrm{n}}}{{\displaystyle 100}}\right]=0 \text{if }\mathrm{n}\le 49$

Again $\frac{1}{2}+\frac{\mathrm{n}}{100}\ge 1 \text{if }\mathrm{n}\ge 50$

$\therefore \left[\frac{{\displaystyle 1}}{{\displaystyle 2}}+\frac{{\displaystyle \mathrm{n}}}{{\displaystyle 100}}\right]=1\text{if}50\le \mathrm{n}\le 99$

The desired sum = 50