u r almost done
for the basis, it is 1 region for 0 lines
for inductive step,
every k th line adds k regions to the sum.
thus we have 1+ sigma(k) regions for k lines
sigma(k)=n(n+1)/2 is readily available
Hi. Im a freshman student.
This question taken from the book by Matoussek and Nestril.
The statement goes:
Can someone please hint me as to how to adress this problem. Answers would be appreciated too but use spoiler tags or different font color so that I may read it after a few attempts by myself.Let us draw n lines in the plane in such a way that no two are parallel and no three intersect in a common point. Prove that the plane is divided into exactly n(n+1)/2 +1 parts by the lines
The only conclusion I have drawn is that the addition of an extra line to a plane containing n lines splits (n+1) regions into two parts.
For example :
# One line on a plane:
Two parts
# Add the (n+1)-th line ( i.e, now there are two lines):
n+1 planes split (i.e, 1+1=2, 2 planes split to form total 4 planes)
#so on...
now what?
Havnt really reached this part.Similarly, consider n planes in the 3-dimensional space in general position (no two are parallel, any three have exactly one point in common, and no four have a common point). What is the number of regions into which these planes partition the space?