If three quantities are in continued proportion; show that the ratio of the first to third is duplicate ratio of first to the second.

Let x, y and z be in continued proportion.

∴ x : y = y : z

$\Rightarrow \dfrac{x}{y} = \dfrac{y}{z} \Rightarrow y^2 = xz$

To prove :

$\dfrac{x}{z} = \dfrac{x^2}{y^2}$

Substituting y² = xz in R.H.S. of above equation we get,

$\dfrac{x^2}{xz} = \dfrac{x}{z}$ = L.H.S.

Hence, proved that ratio of the first to third is duplicate ratio of first to the second.