A moving body travels distance d at uniform speed V1, and travel distance d at uniform speed V2 prove that its average speed is 2V1*V2/V1 + V2

Use the average speed formula.

Average\ Speed = \frac{Total\ Distance}{Total\ time}

The distance now, is 2d.

The time, t_1 for the first part of the journey is given by \frac{d}{v_1}

That for the second part is given by \frac{d}{v_2}

Sub in your equation;

Average\ Speed = \frac{2d}{\frac{d}{v_1}+\frac{d}{v_2}}

Now you can simplify :)

[Thanks ebaines for the other one ;)]