Thickness of Multiple Spherical Shells

The volume of the entire sphere is 
\[\frac{4}{3}\pi (1700)^3.\]
The volume of the smaller sphere inside is 
\[\frac{4}{3}\pi (300)^3.\]
The volume of the shell between these two spheres is 
\[V=\frac{4}{3}\pi (1700)^3-\frac{4}{3}\pi (300)^3=\frac{4}{3}\pi (4886000000).\]
Let $r_0$ be the radius of the first layer. Similarly define $r_{i}$ be the radius of layer $i$, $i=1,2,3, \dots 10$. 
Then
\[\frac{4}{3}\pi r_0^3-\frac{4}{3}\pi (300)^3= \frac{50}{100}V=\frac{50}{100}(\frac{4}{3}\pi (4886000000))\]
\[\Rightarrow r_0^3-(300)^3=\frac{50}{100}(4886000000)=2443000000 \Rightarrow r_0^3= 2470000000 . \]
Now to find $r_1$ we must have
\[\frac{4}{3}\pi r_1^3-\frac{4}{3}\pi r_0^3= \frac{5}{100}\times \frac{1}{2}V=\frac{5}{200}(\frac{4}{3}\pi (4886000000))\]
\[ r_1^3-r_0^3= 244300000 \Rightarrow r_1^3=2470000000+244300000=2714300000.\]
In general 
\[\frac{4}{3}\pi r_{i}^3-\frac{4}{3}\pi r_{i-1}^3= \frac{5}{100}V=\frac{5}{100}(\frac{4}{3}\pi (4886000000))\] \[ r_{i}^3-r_{i-1}^3= 244300000 \Rightarrow r_{i}^3=r_{i-1}^3+244300000.\]
Hence you can find all $r_i$ recursively by the following formula
\[r_0= (2470000000)^{1/3},    r_{i}^3=r_{i-1}^3+244300000,  i=1,2,3, \dots 10.\]