What is the greatest number of parts a plane can be divided into using $n$ infinite straight lines? What about $n$ circles?
Can you generalise this into 3-dimensional space, planes and spheres?
For lines, I got $U_{n+1}=U_n+n,$ with $U_0=1.$ And for circles, I got $U_{n+1}=U_n+2n,$ with $U_0=1$ and $U_1=2.$
Am I right so far?