Question

If q is the mean proportional between p and r prove that `(p^3 + q^3 + r^3)/(p^2q^2r^2) = 1/p^3 + 1/q^3 = 1/r^3`

Answer 1

Since q is the mean proportional between p and r

`q^2 = pr`

L.H.S = `(p^3 + q^3 + r^3)/(p^2q^2r^2)`

`= (p^3 + (pr)q + r^3)/(p^3r^3)`

`= 1/r^3 + q/(p^2r^2) + 1/p^3`

`= 1/r^3 + q/(q^2)^2 + 1/p^3`

`= 1/r^3 + 1/q^3 + 1/p^3`

= R.H.S