Question

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

\[ [\textbf{Hint :}\ \dfrac{a}{b} = \dfrac{b}{c} = k \Rightarrow b = ck \ \textit{and} \ a = ck^{2}\,] \]

Answer 1

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

To prove: $$ad(c^2 + d^2) = c^3(b + d)$$

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. $$\text{L.H.S.} = ad(c^2 + d^2) = (dk^3)(d)((dk)^2 + d^2) = d^2 k^3 (d^2 k^2 + d^2) = d^4 k^3(k^2 + 1)$$
3. $$\text{R.H.S.} = c^3(b + d) = (dk)^3(dk^2 + d) = d^3 k^3 \cdot d(k^2 + 1) = d^4 k^3(k^2 + 1)$$
4. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.