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Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is n(n − 3) 2 - for n ≥ 3 Use the following bullet points and the picture to help write your proof by induction. Break down method: Start with a k+1-gon. Select a vertex and remove it and the k edges attached to it. The resulting figure is a k-gon so you have P(k) diagonals. Add back the vertex and the k edges to form the k+1-gon. Determine how many of these k edges are diagonals (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon. Build up method: Start with a k-gon. It has P(k) diagonals. Add a new vertex and connect it to the k vertices of the k-gon so k edges are added. This creates a k+1-gon. Determine how many of these k edges are diagonals of the k+1-gon (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon.

Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is n(n − 3) 2 - for n ≥ 3 Use the following bullet points and the picture to help write your proof by induction. Break down method: Start with a k+1-gon. Select a vertex and remove it and the k edges attached to it. The resulting figure is a k-gon so you have P(k) diagonals. Add back the vertex and the k edges to form the k+1-gon. Determine how many of these k edges are diagonals (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon. Build up method: Start with a k-gon. It has P(k) diagonals. Add a new vertex and connect it to the k vertices of the k-gon so k edges are added. This creates a k+1-gon. Determine how many of these k edges are diagonals of the k+1-gon (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please write a proof by induction for the following problem

Must have Base Case and show Inductive Step with k+1 and prove true.

Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is n(n − 3) 2 - for n ≥ 3 Use the following bullet points and the picture to help write your proof by induction. Break down method: Start with a k+1-gon. Select a vertex and remove it and the k edges attached to it. The resulting figure is a k-gon so you have P(k) diagonals. Add back the vertex and the k edges to form the k+1-gon. Determine how many of these k edges are diagonals (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon. Build up method: Start with a k-gon. It has P(k) diagonals. Add a new vertex and connect it to the k vertices of the k-gon so k edges are added. This creates a k+1-gon. Determine how many of these k edges are diagonals of the k+1-gon (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon.
Transcribed Image Text:Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is n(n − 3) 2 - for n ≥ 3 Use the following bullet points and the picture to help write your proof by induction. Break down method: Start with a k+1-gon. Select a vertex and remove it and the k edges attached to it. The resulting figure is a k-gon so you have P(k) diagonals. Add back the vertex and the k edges to form the k+1-gon. Determine how many of these k edges are diagonals (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon. Build up method: Start with a k-gon. It has P(k) diagonals. Add a new vertex and connect it to the k vertices of the k-gon so k edges are added. This creates a k+1-gon. Determine how many of these k edges are diagonals of the k+1-gon (or how many are not diagonals). Determine if any of the edges in the k-gon are now diagonals of the k+1-gon but were not in the k-gon.
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