An object moves with speed $$v_1$$, $$v_2$$ and $$v_3$$ along a line segment $$AB$$, $$BC$$ and $$CD$$ respectively as shown in the figure, where $$AB = BC$$ and $$AD = 3AB$$, then the average speed of the object will be:

$$v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}}$$

$$AB = BC = x$$

$$AD = 3AB = 3x$$

$$CD = AD - (AB + BC) = 3x - 2x = x$$

$$t_1 = \frac{x}{v_1}, \quad t_2 = \frac{x}{v_2}, \quad t_3 = \frac{x}{v_3}$$

$$v_{avg} = \frac{x + x + x}{t_1 + t_2 + t_3}$$

$$v_{avg} = \frac{3x}{\frac{x}{v_1} + \frac{x}{v_2} + \frac{x}{v_3}}$$

$$v_{avg} = \frac{3}{\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}}$$

$$v_{avg} = \frac{3v_1v_2v_3}{v_1v_2 + v_2v_3 + v_3v_1}$$