Question

If a, b, c are in G.P., prove that:

$$a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3})=a^3+b^3+c^3$$

Answer 1

a, b and c are in G.P.

$$\therefore b^2=ac$$   .......(1)

$$\text{ LHS }=a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3})$$

$$=\frac{b^2{c}^2}{a}+\frac{a^2{c}^2}{b}+\frac{a^2{b}^2}{c}$$

$$=\frac{\left(ac\right)c^2}{a}+\frac{{\left(b^2\right)}^2}{b}+\frac{a^2\left(ac\right)}{c}[\text{ Using }(1)]$$

$$=a^3+b^3+c^3=\text{ RHS }$$