Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions

Here is the way I usually think about this (it sort of uses induction).

With $0$ lines, there is $1$ region and no intersections of lines.

Each time a line is added and it crosses $k$ other lines it adds $k+1$ regions and $k$ intersections. Another way of looking at this is that for each line and $k$ intersections added, $k+1$ regions are added (the number of added lines and intersections).

Therefore, the number of regions is $1+\text{the number of lines}+\text{the number of intersections}$. With $n$ lines, there are $\binom{n}{2}$ intersections (if no two lines are parallel and no three lines are coincident).

Thus, the number of regions is $\binom{n}{2}+n+1=\frac{n(n-1)}{2}+n+1=\frac{n^2+n+2}{2}$.

Suppose we draw $n$ straight lines on the plane so that every pair of lines intersects (but no $3$ lines intersect at a common point). Into how many regions do these $n$ lines divide the plane?

With $n = 1$ we divide the plane into $2$ regions. With $n = 2$ we have $4$ regions; with $n = 3$ we get $7$ regions. A fourth line will meet the other $3$ lines in $3$ points and so traverse $4$ regions, dividing them into $2$ parts and adding $4$ new regions. In general the $n^{th}$ line will add $n$ new regions:

$$u(1) = 2$$ $$u(2) = 4$$ $$u(3) = 7$$ $$u(4) = 11$$

And so on, where $u(n) =$ number of regions with $n$ lines.

We get the recurrence relationship:

$$u(n+1) = u(n) + (n+1)$$

We get the following chain of equations:

$$u(n) - u(n-1) = n$$ $$u(n-1) - u(n-2) = n-1$$ $$u(n-2) - u(n-3) = n-2$$ $$\vdots$$ $$u(4) - u(3) = 4$$ $$u(3) - u(2) = 3$$ $$u(2) - u(1) = 2$$

Adding these equations, we get: $$u(n) - u(1) = 2 + 3 + 4 + ..... + (n-1) + n$$

All other terms on the left cancel between rows, and we are left with:

$$u(n) = u(1) + 2 + 3 + 4 + \ldots + n$$

We know, $u(1) = 2$

Thus:

$$u(n) = 1 + (1+2+3+4+ \ldots+n)$$ $$\implies u(n) = 1 + \dfrac{n(n+1)}{2}$$ $$\implies u(n) = \dfrac{n^2 + n + 2}{2}$$

So:

$$u(n) = \dfrac{n^2 + n + 2}{2}$$

Remark $\,$ If you allow parallel lines and more than $2$ lines to intersect at a point, the above relation doesn't hold.

The answer then depends on the number of lines intersecting at a point or the number of lines which are parallel to one another.