# geometry – How to find r4/r1?

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)