Use mathematical induction to show that $n$ lines in the plane passing through the same point divide the plane into $2 n$ parts.

Question

Quizplus · Accepted Answer

The Answer is The basis step follows since one line divides the plane into 2 · 1 parts. For the inductive step assume that k lines passing through a point divide the plane into 2k parts. Suppose that we have k + 1 lines. If we take k of these lines, by the inductive hypothesis they divide the plane into 2k parts. Adding the (k+1)st line splits exactly two of these parts in two. Hence these k + 1 concurrent lines split the plane into 2k + 2 = 2 · (k + 1) parts. This completes the proof.

Use Mathematical Induction to Show That nnn Lines in the Plane Passing Through the Same Point Divide

Use Mathematical Induction to Show That $n$ Lines in the Plane Passing Through the Same Point Divide