Question

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

Answer 1

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

To prove: $$(b + c)(b + d) = (c + a)(c + d)$$

Proof:

1. Let $$\frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k$$, which gives $$c = dk$$, $$b = dk^2$$ and $$a = dk^3$$.
2. $$\text{L.H.S.} = (b + c)(b + d) = (dk^2 + dk)(dk^2 + d) = dk(k + 1) \cdot d(k^2 + 1) = d^2 k(k + 1)(k^2 + 1)$$
3. $$\text{R.H.S.} = (c + a)(c + d) = (dk + dk^3)(dk + d) = dk(1 + k^2) \cdot d(k + 1) = d^2 k(k + 1)(k^2 + 1)$$
4. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.