For a real number x, [x] denotes the integral part of x. The value of $[\frac{1}{2}]+[\frac{1}{2}+\frac{1}{100}]+[\frac{1}{2}+\frac{2}{100}]+.....+[\frac{1}{2}+\frac{99}{100}]$ is

50

∵ [x] denotes the integral part of x.

Hence, after term $[\frac{1}{2}+\frac{50}{100}]$, each term will be one. Hence the sum of given series will be 50.