Question

If $\frac{a}{b}=bc=cd,$ prove that $\frac{a}{d}$ is equal to $\frac{a^5+b^2{c}^2+a^3{c}^2}{b^{4}c+d^4+b^{2}c{d}^2}.$

Answer 1

Say $\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=t\Rightarrow a=bt,b=ct,c=dt$

Substituing the above values in the equation

$\frac{a^5+b^2{c}^2+a^3{c}^2}{b^{2}c+d^4+b^{2}c{d}^2}$

Therefore $\frac{{\left(bt\right)}^5+(ct{)}^2{\left(dt\right)}^2+{\left(bt\right)}^3{\left(dt\right)}^2}{{\left(ct\right)}^{2}c+d^4+{\left(ct\right)}^{2}c{d}^2}=\frac{a}{d}$