Hello,
Let's assume i the number of lines,
u(i) the number of regions:
[tex]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 }[/tex]
