Help proving a recursive formula involving planes and lines Suppose that $n$ lines are drawn on a plane in such a way that no lines are parallel and no three of them intersect at a point. Let $r_n$ be the number of regions in the plane is divided into after drawing the $n$ lines.
a) Find a recursive formula for the sequence {$r_n$}, $n\geq0$. Prove (without using induction) that your formula is correct.
b) Find a close formula for the generating function of the sequence {$r_n$}.
So for part a), I've found the recursion formula by drawing out the plane and the first few lines.
Since I got $n=0, r_0=1$ ; $n=1, r_1=2$ ; $n=2, r_2=4$ ; $n=3, r_3=7$ ; $n=4, r_4=11$
I'm saying the recursion formula is $r_n=r_{n-1}+n$. I'm not sure how I can prove this without induction. I think I need some way to show that adding an $n$th line will add $n$ regions but I have no idea how to go about this.
For b) I set $R(x)=r_0+r_1x+r_2x^2+...+r_nx^n$. Then,
$$R(x)=r_0+r_1x+...+r_nx^n$$
$$=r_0+r_1x+\sum_{k\geq0}(r_{k-1}+k)x^k$$
$$=r_0+r_1x+\sum_{k\geq0}r_{k-1}x^k+\sum_{k\geq0}kx^k$$
$$=r_0+r_1+(r_1x^2+r_2x^3+...)+(2x^2+3x^3...)$$
$$=r_0+r_1+xF(x)+\sum_{k\geq0}kx^k$$
Solving for $R(x)$ gives me
$$R(x)=\frac{1+2x+\sum_{k\geq0}x^k}{1-x}$$
I feel like I'm missing something at this step, since I'm not sure what to do with the $$\sum_{k\geq0}kx^k.$$ I thought about using the identity $$\sum_{k\geq0}x^k=\frac{1}{1-x}$$ but I don't know if I'm allowed to do that since we $k$ as a coefficient, so it's not like a constant I can just pull out. 
Any thoughts on how I can keep going in the right direction for either a) or b) would be really appreciated!

if L intersect whit the L',divides two neighbor region to four region ,so it adds tow region.we count each region twice except two region at infinity.so it adds $(2(n-1)+2)/2=n$region  
