There are n lines drawn in a plane such that no 2 lines are parallel and no 3 lines are concurrent. If the plane is then divided into an regions prove that $$a_1=2,a_2=4,a_n=a_{n-1}+n \; \mathrm{for} \; n\ge2$$ Find the generating function for this sequence
Try: Starting from $a_0 = 1$, the sequence will be: $1,2,4,7,11,16,22,29,37,46,56,..... $
Therefore the generating function will be of the form $g(x) = 1+2x+4x^2+7x^3+...$
Suggested steps.
Begin with $$ g(x) = \sum_{n=0}^\infty a(n) x^n = 1 + \sum_{n=1}^\infty a(n)x^n $$ as the OP suggests.
Next, in terms of $g$, what is $$ \sum_{n=1}^\infty a(n-1)x^n $$ What is $$ \sum_{n=1}^\infty n x^n $$
Put these three results together (using your recurrence $a(n)=a(n-1)+n$) to get an equation satisfied by $g$.
Solve it, to determine what $g$ actually is.