If $a, x, b$are in H.P. and $a, y, z, b$ are in G.P., then the value of $\frac{yz}{x\left(y^3+z^3\right)}$ is

$x=\frac{2ab}{a+h}$

Let common ratio of G.P. $a, y, z, b$

be $r,$ then$r^3 = b/a$ and

$\begin{array}{l}y=ar,z=b/r\\ \therefore yz=ab\end{array}$

and $y^3+z^3=a^3{r}^3+\frac{b^3}{r^3}$

$\begin{array}{l}=a^3\left(\frac{b}{a}\right)+b^3\left(\frac{a}{b}\right)\\ =ab(a+b)\\ \Rightarrow x\left(y^3+z^3\right)=\frac{2ab}{a+b}\left(ab\right)(a+b)=2a^2{b}^2\\ \therefore \frac{yz}{x\left(y^3+z^3\right)}=\frac{ab}{2(ab{)}^2}=\frac{1}{2ab}\end{array}$