Mathematical Induction Question?

Question

For each positive interger n, define R(n) to be number of regions into which n lines, no two are parallel and no three of which intersect at the same point, divide the line.

Determine a forumla for R(n) and prove by induction it is true for all positive n integers...

I got a couple formulas but I don't know how to do the induction part...it's like...r(n) = n(n+1)/2 + 1 or r(n) = r(n-1) + n or r(n+1) = r(n) + n + 1...

which one do I use? how do I do the induction? thanks

Is there anyway I can contact you? Or you being able to explain the steps to do it? I'm really confused on this problem, been trying to figure it out for the past 2 days.

Answer 1

Your formulas are correct, you have both R(n) = n(n + 1)/2 + 1 and R(n+1) = R(n) + n + 1. Let us start with R(0) = 1, which we understand to mean that 0 lines leave the plane intact, so 1 region.

Now R(1) = 2 = R(0) + 1 or R(1) = [(0)*(0 + 1)/2 +1] +1 = 2, so the induction is anchored,

Assume that R(n) = n(n + 1)/2 + 1. Now the conditions (that no two lines are parallel and no three hit the same point) ensure that an (n+1)-st line cuts all previous n lines. But a new line cutting n old lines divides the plane into n+1 new regions. Therefore R(n+1) = R(n) + (n+1) = [n(n+1)/2 + 1] + (n+1) = [n(n+1)/2 + (n+1)] + 1 = [(n^2 + n + 2n + 2)/2 + 1 = (n+1)(n+2)/2 + 1. This completes the proof.

Answer 2

it is impossible for me to type it out on yahoo answers even though i know how to do