Answer
$\mu_2 = 0.44~\mu_1$
$\mu_3 = 0.19~\mu_1$
$\mu_4 = 0.084~\mu_1$
Work Step by Step
Let $\mu_1$ be the mass per unit of the lowest string;
$f_1 = \frac{1}{2L}~\sqrt{\frac{F_T}{\mu_1}}$
We can find the mass per unit $\mu_2$ of the second string in terms of $\mu_1$ as:
$f_2 = 1.5~f_1$
$\frac{1}{2L}~\sqrt{\frac{F_T}{\mu_2}} = (1.5)~\frac{1}{2L}~\sqrt{\frac{F_T}{\mu_1}}$
$\frac{1}{\sqrt{\mu_2}} = (1.5)(\frac{1}{\sqrt{\mu_2}})$
$\mu_2 = \frac{\mu_1}{(1.5)^2}$
$\mu_2 = 0.44~\mu_1$
Similarly:
$\mu_3 = 0.44~\mu_2$
$\mu_3 = 0.44~(0.44~\mu_1)$
$\mu_3 = 0.19~\mu_1$
Similarly:
$\mu_4 = 0.44~\mu_3$
$\mu_4 = 0.44~(0.19~\mu_1)$
$\mu_4 = 0.084~\mu_1$
