Why do this problem?
Here you have a result that must be proved for all positive whole
numbers $n$ which suggests that proof by induction is
the obvious method to try. The result involves ratios of Fibonacci
numbers so it is certainly non-standard. Even so the algebra
involved is elementary and so the problem is a good exercise in
using proof by induction. It also links apparently unrelated
mathematical topics such as Fibonacci numbers and mediants and the
geometry of Ford Circles
Fibonacci numbers should be part of the repertoire of every
mathematician as they are useful in many appllications in higher
How about using the 'Moore' method here? In other words tell your
students that the class must work out the solution together, that
you will stand at the back of the room and you will observe and
just ask a few questions. If you have not taught in this way before
explain why you are choosing this approach and how you believe it
will benefit them.
Invite the students to decide between them who will record on the
board what needs to be recorded. They should first decide on what
method to use and exactly what is involved in using that method of
proof. Then it works well for the learners to work individually to
complete the proof before someone writes it up on the board for
discussion. When the class gets stuck, or when they think that they
have completed the proof successfully, then they need to check each
step carefully and everyone should agree that it is correct before
The teacher will need to exercise restraint so as not to make
suggestions or point out mistakes even if there are periods when
the class is stuck and does not seem to be making progress. This is
an exercise in learning how to
and this method of teaching can be very effective once
the students realise that they have to think for themselves and the
teacher is not going to lead them. If absolutely necessary the
teacher should ask a question that prompts the learners to consider
issues that are conducive to finding a way forward or to spotting
their own mistakes or omissions.
Before teaching in this way it is essential for the teacher to have
a good appreciation of all the alternative methods of solution and
to have worked through the question themselves so that they know
the likely pitfalls.
What questions should we be asking ourselves at this stage? [Keep
asking this question to encourage the learners to ask themselves
the key questions. The teacher should only ask the following
questions as a last resort if the class are not making
Have we understood what the question is asking us to do?
Have we investigated the result for some small values of $n$ and
checked them numerically?
Why does the absolute value appear here?
What values of $n$ are we asked to prove the result for and what
method of proof does this suggest?
Exactly what do we have to do to give a proof by induction?
In proving that the truth of the result for $n=k+1$ follows from
the truth of the result for $n=k$ what formula should we work
In trying to solve this problem, have we used all the information
we were given?
If $a+b=c$ then what do we know about $c-b$?
Try the problems Farey
Try the problem Farey