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}{3} = frac{A_4}{4}.)Let $r_1$ denote the radius of the smallest circle, and let $r_4$ denote the radius of the largest circle. Find $frac{r_4}{r_1}.$

1Latex code:

(asy)

unitsize(1 cm);

pair() O;

real() r;

O1 = (0,0);

O(2) = (0.1,0.2);

O(3) = (-0.2,-0.1);

O(4) = (0.1,-0.3);

r1 = 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(O1,r1),lightred); draw(Circle(O1,r1));label(“$A_1$“, O1);

(/asy)