Given that
$\begin{aligned}
&(1+x)^{15}=a_0+a_1 x+a_2 x^2+\ldots+a_{15} x^{15} \\
& \Rightarrow \quad{ }^{15} C_0+{ }^{15} C_1 x+{ }^{15} C_2 x^2+\ldots+{ }^{15} C_{15} x^{15} \\
&=a_0+a_1 x+a_2 x^2+\ldots+a_{15} x^{15}
\end{aligned}$
Equating the coefficient of various powers of $x$, we get
$\begin{aligned}
& a_0={ }^{15} C_0, a_1={ }^{15} C_1, a_2={ }^{15} C_2, \ldots, a_{15}={ }^{15} C_{15} \\
\therefore & \sum_{r=1}^{15} r \frac{a_r}{a_{r-1}}=\sum_{r=1}^{15} r \frac{{ }^{15} C_r}{{ }^{15} C_{r-1}} \\
= & \sum_{r=1}^{15} r \frac{\frac{15 !}{r !(15-r) !}}{\frac{15 !}{(r-1) !(15-r+1) !}} \\
= & \sum_{r=1}^{15} \frac{(r-1) !(15-r+1) !}{r !(15-r) !} \\
= & \sum_{r=1}^{15} 15-r+1=15+14+13+\ldots+2+1 \\
= & \frac{15(15+1)}{2}=120
\end{aligned}$