Question

Into how many segments do n points divide a given line segment?

Answer 1

n points would divide a given line into n+1 segments.

Just assume that you have a line and you added a point to that line, this point would divide your line into two segments.
If you add two points to a line, these two points would divide your line into three segments.
So the number of segments is always greater that the number of points by 1.