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### Why do this problem?

This
problem provides a context for genuine discovery and student
directed research/investigation. It is an ideal situation for
students to work out how to use ICT to speed up the calculations
and support them in their quest to rediscover Litov's
Theorem.

### Possible approach

"I found this problem called Litov's Mean Value Theorem. I'm hoping
we can work out what the theorem is. We can start with any two
numbers, say $8$ and $2$. These are the start of a sequence. The
rule is that the next number in the sequence is the average of the
last two numbers. So what comes next? Why? And then what?...

Invite students to choose their own pair of starting numbers, to
calculate the sequence and find its limit. Students could use
calculators for this activity. Giving students free choice can
result in a lot of information being collected in a short space of
time. While this is going on and the results are appearing on the
blackboard, ask some students to think about how these calculations
could be done on a spreadsheet.

Bring the class together and ask for observations, comments,
suggestions and predictions. Demonstrate the use of a spreadsheet
for testing these predictions quickly. The use of the computer
makes it possible to operate at a new level and the computer shows
the limiting process clearly.

"Given all that information would anyone like to check a result or
predict what will happen to any pair of numbers?"

There's a chance to discuss whether these sequences will ever
actually reach their limits.

Students could test their hypotheses working on paper, or everyone
could be given access to spreadsheets. When students are convinced
that they know how to find these limits, challenge them to suggest
some reasons why the limits behave as they do.

Students could then move on to working on these:

What would happen if sequences were generated from three initial
values by:

averaging the last three numbers
(i.e. $(a+b+c)/3$)?

adding the last three numbers and dividing by 2 (i.e.
$(a+b+c)/2$)?

### Key questions

Can you tell where these numbers are heading?

Does it matter if I swap the two starting numbers around?

What do these long decimals mean? How big is that number, roughly?

### Possible extension

What happens when you have $n$ start numbers and the rule for
working out the next number changes to finding the average of the
last $n$ numbers?

### Possible support

This problem is a good context for work on organisation skills and
calculator competence with opportunities for making conjectures,
and refining conjectures.

Laurinda Brown (1983) wrote about using this
problem in the classroom: in Mathematics...with a Micro 1,
pp.22-25, Waddingham, Jo (ed), Bristol, County of Avon, Resources
for Learning Development Unit. The lesson notes above are adapted
from her descriptions of using the problem.