Category Archives: Abstract algebra

December 31, 2009, 5:30 am

How to convert a "backwards" proof into a "forwards" proof

Dave Richeson at Division By Zero wrote recently about a “proof technique” for proving equalities or inequalities that is far too common: Starting with the equality to be proven and working backwards to end at a true statement. This is a technique that is almost a valid way to prove things, but it contains — and engenders — serious flaws in logic and the concept of proof that can really get students into trouble later on.

I left a comment there that spells out my¬† feelings about why this technique is bad. What I wanted to focus on here is something I also mentioned in the comments, which was that it’s so easy to take a “backwards” proof and turn it into a “forwards” one that there’s no reason not to do it.

Take the following problem: Prove that, for all natural numbers \(n\),

\(1 + 2 + 2^2 + \cdots + 2^n = 2^{n+1} – 1\)

This is a standard exercise using mathematical…

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