Four circles are drawn. Let $A_1,$ $A_2,$ $A_3,$ $A_4$ be the areas of the regions, so $A_1$ is the area inside the smallest circle, $A_2$ is the area outside the smallest circle and inside the second-smallest circle, and so on. The areas satisfy [A_1 = \frac{A_2}{2} = \frac{A_3}{4} = \frac{A_4}{5}.]Let $r_1$ denote the radius of the smallest circle, and let $r_4$ denote the radius of the largest circle. Find $\frac{r_4}{r_1}.$

[asy] unitsize(1 cm);

pair[] O; real[] r;

O[1] = (0,0); O[2] = (0.1,0.2); O[3] = (-0.2,-0.1); O[4] = (0.1,-0.3);

r[1] = 1; r[2] = 1.5; r[3] = 2; r[4] = 2.5;

fill(Circle(O[4],r[4]),lightblue); draw(Circle(O[4],r[4])); label("$A_4$", (1.8,-1.5)); fill(Circle(O[3],r[3]),lightgreen); draw(Circle(O[3],r[3]));label("$A_3$", (-1.3,-1.3)); fill(Circle(O[2],r[2]),yellow); draw(Circle(O[2],r[2]));label("$A_2$", (1,1)); fill(Circle(O[1],r[1]),lightred); draw(Circle(O[1],r[1]));label("$A_1$", O[1]); [/asy]