Question

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

Answer 1

Let x y and z be the three quantities which are in continued proportion.
Then
$x:y::y:z$
$\to y^2=xz.$...(1)
now we have to prove that
$x:z=x^2:y^2$
i.e.
we need to prove that
$x{y}^2=x^2\times z$

L. H. S
$x{y}^2=x\times \left(xz\right)=x^2\times z$ = R. H. S
Hence proved