Proving a Polygon's Diagonal Formula with Induction | Step-by-Step Guide

Thread starter Nikitin
Start date Oct 31, 2011
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Induction

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SUMMARY

The discussion centers on proving the formula for the number of diagonals in a polygon with n angles, expressed as Mn = (n-3)n/2, using mathematical induction. The participants clarify the induction hypothesis, which states that for a polygon with k angles, the number of diagonals is Mk = (k - 3)k/2. They demonstrate that the formula holds for n=3 (a triangle) and derive the recursive relationship Mn+1 = Mn + n - 1 to establish the proof for n+1. The conversation emphasizes the importance of correctly structuring the induction steps and understanding the relationship between Mk and Mk+1.

PREREQUISITES

Understanding of mathematical induction principles
Familiarity with polygon properties and diagonal calculations
Knowledge of recursive formulas in mathematics
Ability to manipulate algebraic expressions

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Study mathematical induction proofs in detail
Learn about recursive relationships in combinatorial mathematics
Explore properties of polygons and their diagonals
Practice constructing induction proofs with various mathematical statements

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Students studying geometry, mathematics educators teaching induction proofs, and anyone interested in combinatorial mathematics and polygon properties.

Oct 31, 2011

Ibix

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Mark44 said:

I disagree that what needs to be done is "rearrange" M_k + D to look like M_k+1. I might be misinterpreting what you mean by "rearrange" here, but my problem with nikotin's work is that it seems like all he is doing is replacing n by n + 1 in the formula, which is not how an induction proof is proved. If that's not what he's doing, then his work needs to be clearer.

I was getting at what you were getting at: that if you know how to transform the number of diagonals for a k-gon into that for a (k+1)-gon, and if that transform maps the proposed number of diagonals for a k-gon to the proposed number of diagonals for a (k+1)-gon then the inductive hypothesis is satisfied. All that remains to be done is to confirm that the proposed number of diagonals is correct for one case.

OP: I sometimes find a counter-example helpful. I propose that the number of diagonals is much simpler: M'_n=n. The inductive hypothesis would have it that if this is true for some n=k then M'_k+1 (=k+1) should be equal to M'_k+k-1 (=2k-1). Since this statement is not true, my idea is not consistent with the inductive hypothesis and so fails.

Try again: M''_n=M_n+3. This does satisfy the inductive hypothesis (easy to verify since I've stolen the correct solution and broken it). So, if it were true for some n=k, then it would be true for all n. However, it is not true for a concrete value of n, say n=3, so fails.