If a, b, c are in continued proportion, prove that :

(i) $\dfrac{a + b}{b + c} = \dfrac{a^{2}(b - c)}{b^{2}(a - b)}$

(ii) $\dfrac{a + b + c}{a - b + c} = \dfrac{(a + b + c)^{2}}{(a^{2} + b^{2} + c^{2})}$

(iii) $\dfrac{a^{2} + ab + b^{2}}{b^{2} + bc + c^{2}} = \dfrac{a}{c}$

(iv) (a + b + c)(a − b + c) = (a² + b² + c²)

(v) a²b²c²(a⁻³ + b⁻³ + c⁻³) = (a³ + b³ + c³)

(vi) ad(c² + d²) = c³(b + d)

If x, y and z are in continued proportion, prove that :

$\dfrac{x}{y^2.z^2} + \dfrac{y}{z^2.x^2} + \dfrac{z}{x^2.y^2} = \dfrac{1}{x^3} + \dfrac{1}{y^3} + \dfrac{1}{z^3}$

If a, b, c, d are in continued proportion, prove that :

(a + b)(b + c) - (a + c)(b + d) = (b - c)²

If a, b, c, d are in continued proportion, prove that :

(a² − b²)(c² − d²) = (b² − c²)²