I got this problem from here; the directions instruct to solve it by induction. I would appreciate it if someone could give me a hint on this problem.
One of the aspects which makes this problem difficult for me is that the following statement is not obvious (if it is even true) for me: "If you have arranged $n$ lines to give you the maximum number of regions possible with $n$ lines, the maximum number of regions possible with $n+1$ lines can be achieved by adding a line to the previous $n$ line configuration"
Although I haven't found a counterexample to the statement, I did find that by using $n=2$ and $n=3$ you can make the sequences (of number of regions) $(3, 6)$ and $(4, 6)$.
By sketching for $n = 1, 2, 3, 4$ I got $2, 4, 6, 10$ which is a nice sequence, but unfortunately for $n = 5$ the maximum I could make was $14$, not $16$.
I of course also noticed that if you make a line cutting through $k$ regions, you increase the number of regions by $k$.
Best Answer
You must have been doing something wrong drawing your 5 lines. Just draw a pentagram with extended lines and count clockwise so you're sure you didn't leave anything out:
But your sequence is flawed in other points, too, as Robert Israel pointed out; try ironing that using a similar principle. When you have a guess (or when you're feeling like confirming a hypothesis), it's a matter of proving that. Try thinking of cutting existing regions, and how many of them a new line that crosses each line so far will encounter.
Another possible hint is to see what happens if you go in reverse: starting with my picture, try removing one line and counting again. Then removing another. I'll leave you with that, the problem teaches you more if you do the main part of it yourself.