Question

If $$a, b, c$$ are in continued proportion, prove that $$a^2b^2c^2(a^{-3} + b^{-3} + c^{-3}) = (a^3 + b^3 + c^3)$$.

Answer 1

Given: $$a, b, c$$ are in continued proportion.

To prove: $$a^2b^2c^2(a^{-3} + b^{-3} + c^{-3}) = (a^3 + b^3 + c^3)$$

Proof:

1. Since $$a, b, c$$ are in continued proportion, $$b^2 = ac$$.
2. $$\text{L.H.S.} = a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = \frac{b^2 c^2}{a} + \frac{a^2 c^2}{b} + \frac{a^2 b^2}{c}$$
3. $$\frac{b^2 c^2}{a} = \frac{(ac)c^2}{a} = c^3$$ [$$\because b^2 = ac$$]
4. $$\frac{a^2 c^2}{b} = \frac{(ac)^2}{b} = \frac{(b^2)^2}{b} = b^3$$ [$$\because ac = b^2$$]
5. $$\frac{a^2 b^2}{c} = \frac{a^2(ac)}{c} = a^3$$ [$$\because b^2 = ac$$]
6. $$\text{L.H.S.} = c^3 + b^3 + a^3 = a^3 + b^3 + c^3 = \text{R.H.S.}$$

Hence proved.