Mathematical induction… or not?

Lately I’ve been spending some time investigating the teaching of mathematical induction, and in particular questions which require the proof of a divisibility property. Here’s two:

  1. Prove that for all integers n\ge 0, 7^{n+2}+8^{2n+1} is divisible by 57.
  2. Prove that for all integers n\ge 1, 27^n-26n-1 is divisible by 676.

Interestingly enough, for many such problems – and almost all discrete mathematics texts will include, if not these actual problems, then several very similar to them – induction is not necessarily the simplest method of proof.

Let’s look at the first example. I shall give three proofs: two using induction, and one not. For the induction proofs, the base step is trivial: 7^2+8=57. Now for the inductive step:

  • Assume 7^{k+2}+8^{2k+1}=57N for some N. Replace k with k+1. Then:7^{k+3}+8^{2k+3}=7.7^{k+2}+64.8^{2k+1}\qquad =7.7^{k+2}+(7+57).8^{2k+1}\qquad = 7(7^{k+2}+8^{2k+1})+57.8^{2k+1}\qquad =7.57N+57.8^{2k+1} by the inductive hypothesis\qquad =57(7N+8^{2k+1})
  • We use the fact that for a function f(n) if f(k) is divisible by 57, and if f(k+1)-rf(k) is dvisible by 57 for some r relatively prime to 57, then f(k+1) is divisible by 57.
    Here f(n)=7^{n+2}+8^{2n+1} and if we use r=7, then

Now for a proof not using induction, but a tiny bit of number theory. First, write






How simple is that?

Now for the second example. In fact, it is very easy to prove that (a+1)^n-an-1 is divisible by a^2: expand (a+1)^n using the binomial theorem, and remove the last two terms; the rest will consist of powers of a the lowest of which is a^2. The result follows.

I wonder, of the vast number of induction exercises given to students, how many problems are in fact far more easily proved by other means?

About these ads

One comment

  1. mvngu

    Possible typo?

    > Here’s two:
    Here are two:

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


Get every new post delivered to your Inbox.

Join 44 other followers

%d bloggers like this: