I have an excessive amount of inequalities questions that I dont understand.
Here is another one
Thank you so much
Like Rui said, these types of induction questions are uncommon even in the Ext2 course. They require you to consider what actually happens in the geometric conditions provided.
Here's my working for it.
Firstly, to create the "greatest" number of regions, none of the lines can be parallel (or the same line).
We want to somehow create a link between the number of regions and the expression given. To do this we want to establish a pattern. This can be done by simply drawing and seeing what happens to the number of regions for n amount of lines.
We get:
n=1 makes 2
n=2 makes 4
n=3 makes 7
n=4 makes 11
As we can see, each new line seems to add the amount of regions as the number of line it is, starting with 2 regions for 1 line. We can turn this into a pattern. The number of regions for n lines is 1 + (1 + 2 + 3... + n). Using the sum of an arithmetic series, this can be simplified to 1 + n(n+1)/2 = (n^2 + n + 2)/2. We got the expression given! So we're on the right track.
Now realise we haven't actually proven this expression for n is equal to the number of regions for n lines. we've just noticed a pattern for the first few cases. This is where the induction comes in. For case n=1, there are two regions. (1+1+2)/2 + 2. Proven for initial case.
Assume that for k lines, there are (k^2+k+2)/2 regions.
For the case for k+1 lines, we're trying to prove there are [(k+1)^2+k+1+2)/2 = (k^2+3k+4)/2 regions.
Now what happens when we have k+1 lines? We added an extra line in. As mentioned before, it seems to add k+1 regions. How do we prove this? Notice that the added line will cut k lines, since it isn't parallel to any of them. As it cuts each line, it will split each region the line is in, into two, essentially adding another region, however adding that line itself split the overall circle in two, which adds another region. This means that one region is added, plus one region for every line it crosses. That's k+1 regions added! So for the case n=k+1, the number of regions is equal to the number of regions in the case n=k and then with k+1 regions added.
That's (k^2+k+2)/2 + k + 1 = (k^2+3k+4)/2
True for n=k+1 when true for n=k
True for n=1
Proven by induction
I hope my working wasn't too hard to follow. You can see why they don't ask these questions anymore.