If x, y, z are three sum of money such that y is simple interest on x and z is simple interest on y for same time and at same rate of interest, n what is relation between x, y, z we have = ?

Let the time is 't' years and the rate of interest is R%
According to the question,
$$\eqalign{ & {\bf{Case(I):}} \cr & y = \frac{{x \times {\text{R}} \times {\text{t}}}}{{100}}.....(i) \cr & {\bf{Case(II):}} \cr & z = \frac{{y \times {\text{R}} \times {\text{t}}}}{{100}}.....(ii) \cr} $$
By dividing equation (i) by equation (ii)
$$\eqalign{ & \frac{y}{z} = \frac{{x \times {\text{R}} \times {\text{t}}}}{{y \times {\text{R}} \times {\text{t}}}} \cr & \Rightarrow {y^2} = zx \cr} $$