A person travels x distance with velocity v 1 and then x distance with velocity v 2 in the same direction.…

The time $\left(t_1\right)$ taken by the person to travel distance $x$ with velocity $v_1$ is given by

$t_1=\frac{x}{v_1} ...\left(1\right)$

The time $\left(t_2\right)$ taken by the person to travel distance $x$ with velocity $v_2$ is given by

$t_2=\frac{x}{v_2} ...\left(2\right)$

The formula to calculate the average speed $\left(v\right)$ of the person is given by

$v=\frac{2x}{t_1+t_2} ...\left(3\right)$

Substitute the expressions from equations (1) and (2) into equation (3) to obtain the required relation.

$\begin{array}{rcl}v& =& \frac{2x}{\left(\frac{x}{v_1}+\frac{x}{v_2}\right)}\\ & \Rightarrow & \frac{x}{v_1}+\frac{x}{v_2}=\frac{2x}{v}\\ & \Rightarrow & \frac{2}{v}=\frac{1}{v_1}+\frac{1}{v_2}\end{array}$