Question

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

Answer 1

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

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

Proof:

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

Hence proved.