Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is 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.
Theorem: The number of diagonals of a regular n- Define P (n) = n(n-3) 2 gon is 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.
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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Question
Please write a proof by induction for the following problem
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6f18ab9-3892-49f5-91ed-269d155e1486%2F55bfe236-2a0a-4883-a498-108b7982c750%2Fn9w622h_processed.png&w=3840&q=75)
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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