Integer Sequences and the Need for Proof

This post over at Mr. Honner on unstated assumptions reminded me to share a great visual that shows some sequences are not what they seem:

patterns1

Of course, the pattern that jumps out is:

2, 4, 8, 16, 32, 64, …

Geometric Sequence!  Exponential function!  2^(n-1).  HOWEVA,

patterns2

I have yet to bust this out in my classroom, but I envision it being my answer the question “Why do we need proofs?”  Without proof or a counterexample, it is awfully easy to assume that a pattern exists when it really doesn’t.  That’s why we need proof.  Without being able to logically show something is true or false, you can fall into the traps of assumption.

I think I like the idea of setting up students to fail here. Just give them the question and let them try to find the pattern. Of course, their assumption will be that they can find a pattern and I’m sure most would be satisfied that 2^n works with n=1, 2, 3, 4, so it must work with all n.  As Lee Corso would say, “Not so fast my friend.”

Update

patterns5Thanks to commenter Graeme McRae for pointing out this sequence on OEIS (Online Encyclopedia of Integer Sequences).  The sequence can also be defined as the sum of the first five terms of the nth row of Pascal’s triangle:

Update #2

Andy Huynh notes that this sequence should be specified as the maximum number of regions created by putting n points on a circle and connecting them with chords.  As I mentioned in my comment, I think this observation again highlights the need to be able to prove something.  Andy notes that if you evenly space the points when n = 6 (create a regular hexagon), then you will end up with only 30 regions, not 31.  In fact, that’s what I did when I first did this problem which was shared at NCTM by Jen Szydlik.

 patterns4