Question

If n lines are drawn in a plane, how many regions do they separate the plane into? induction

Answer 1

Hello,

Let's assume i the number of lines,
 u(i) the number of regions:

$u_0=1\\\\ u_1=2=1+1=1+u_0\\\\ u_2=4=2+u_1\\\\ u_3=7=3+u_2\\\\ u_4=11=4+u_3\\\\ u_5=16=5+u_4\\\\ ...\\\\ \boxed{u_{n+1}=(n+1)+u_{n}}\\\\ u_{n+1}=(n+1)+n+(n-1)+...+1+u_0\\\\ \boxed{u_{n+1}= \dfrac{(n+1)*(n+2)}{2}+1 }$