\(\begin{array}{l}
\frac{1}{{1.2}} + \frac{1}{{3.4}} + ..... + \frac{1}{{99.100}} = \frac{1}{{51}} + \frac{1}{{52}} + ..... + \frac{1}{{100}}\\
VT = \frac{1}{{1.2}} + \frac{1}{{3.4}} + ..... + \frac{1}{{99.100}}\\
= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + .... + \frac{1}{{99}} - \frac{1}{{100}}\\
= \left( {1 + \frac{1}{3} + \frac{1}{5} + ..... + \frac{1}{{99}}} \right) - \left( {\frac{1}{2} + \frac{1}{4} + .... + \frac{1}{{100}}} \right)\\
= \left( {1 + \frac{1}{3} + \frac{1}{5} + ..... + \frac{1}{{99}}} \right) + \left( {\frac{1}{2} + \frac{1}{4} + .... + \frac{1}{{100}}} \right) - 2\left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + .... + \frac{1}{{100}}} \right)\\
= \left( {1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{{99}} + \frac{1}{{100}}} \right) - \left( {1 + \frac{1}{2} + \frac{1}{3} + ..... + \frac{1}{{50}}} \right)\\
= 1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{{99}} + \frac{1}{{100}} - 1 - \frac{1}{2} - \frac{1}{3} - ..... - \frac{1}{{50}}\\
= \frac{1}{{51}} + \frac{1}{{52}} + ... + \frac{1}{{100}}.
\end{array}\)
