Today’s post is a little bit off the main topic of this series of posts… but I wanted to give some pedagogical thoughts on yesterday’s post concerning the following proof by induction.
Theorem: If is a positive integer, then
is a multiple of 4.
Proof. By induction on .
:
, which is clearly a multiple of 4.
: Assume that
is a multiple of 4, so that
, where
is an integer. We can also write this as
.
. We wish to show that
is equal to
for some (different) integer
. To do this, notice that
by the induction hypothesis
.
So if we let , then
, where
is an integer because
is also an integer.
My primary observation is that even very strong math students tend to have a weak spot when it comes to simplifying exponential expressions (as opposed to polynomial expressions). For example, I find that even very good math students can struggle through the logic of this sequence of equalities:
.
The first step is using the main stumbling block. Students who are completely comfortable with simplifying as
can be perplexed by simplifying
as
. I attribute this to lack of practice with this kind of simplification in lower grade levels.
Here’s another algoebraic stumbling block that I’ve often seen: at the beginning of the case, some students will make the following mistake:
.
Because these students end with a multiple of 4, they fail to notice that the second equality is incorrect since
.
Again, I attribute this to lack of practice with simplifying exponential expressions in lower grade levels… as well as being a little bit over-excited upon seeing and wishing to use the induction hypothesis as soon as possible.
One thought on “Proving theorems and special cases (Part 5): Mathematical induction”