Question

Mathematical induction (MI) problem #2
Prove, by MI, that the maximum possible number of regions that a plane can be divided by n lines is \[\frac{1}{2}n(n+1)+1\]

Answer 1

So, first, I should let R(n) be the max. possible number of regions that a plane is divided by n lines; and
p(n) be the statement \[R(n) = \frac{1}{2}n(n+1)+1\]
But how can I tell R(1) =2?

Answer 2

Even though R(1)=2 is the truth.... Should I draw a diagram?

Answer 3

R(1)=2 seems pretty obvious. Kinda depends how rigorous you have to be, I guess. My first thought of how to prove it really rigorously is the trichotomy law, but my guess is that you should be okay with something simple, or just handwaving it altogether.

Answer 4

Actually, it's an example from my book. Before really showing the proof, there is an analysis part, which finally comes up a conclusion that R(k+1) = R (k) + (k+1)
I think I understand it. But when I have to come up with it on my own, that 's another situation.
Back to the question, since there is a word 'prove' in the question, it seems that I really need to prove it, but I just don't know how...

Answer 5

Here's the 'analysis part' I mentioned:
Let R(n) be the max. possible number of regions that a plane is divided by n lines. Obviously, R(1) =2 and R(2) = 4. Suppose k lines have already been drawn on the plane. As illustrate in Fig. 1.2 and Fig. 1.3, the (k+1)th line can increase a max. of (k+1) more regions when it does not pass any intersections of the k lines.

Captions of Fig.1.2: R(3) = 4+3 =7
Captions of Fig.1.3: R(4) = 7+4 =11