Question

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

Answer 1

$\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k$, then

$a=bk=c{k}^2=d{k}^3\text{and}b=ck=d{k}^2\text{and}c=dk$

L.H.S. $=\frac{a}{b+d}=\frac{d{k}^3}{d{k}^2+d}=\frac{k^3}{k^2+1}$

R.H.S. $=\frac{c^3}{c^{2}d+d^3}=\frac{d^3{k}^3}{d^2{k}^2\cdot d+d^3}=\frac{k^3}{k^2+1}$

$\therefore$ L.H.S. $=$ R.H.S.