Question

If x, y, z are in continued proportion, prove that: `(x + y)^2/(y + z)^2 = x/z`.

Answer 1

x, y, z are in continued proportion

Let `x/y = y/z = k`

Then y = kz

x = yk

= kz × k

= k²z

Now L.H.S.

= `(x + y)^2/(y + z)^2`

= `(k^2 z + kz)^2/(kz + z)^2`

= `{kz(k + 1)}^2/{z(k + 1)}^2`

= `(k^2z^2(k + 1)^2)/(z^2(k + 1)^2)`

= k²

R.H.S. = `x/z`

= `(k^2z)/z`

= k²

∴ L.H.S. = R.H.S.