# Litov's Mean Value Theorem

*Litov's Mean Value Theorem printable sheet*

**Start with two numbers, say 8 and 2.**

Let's generate a sequence where the next number is the mean of the previous two numbers.

So the next number is half of $(8 + 2)$, and the sequence continues: $8, 2, 5$

The next number is half of $(2 + 5)$, and the sequence continues: $8, 2, 5, 3.5$**What would happen if you continued this process indefinitely?**

Choose a few pairs of starting numbers and repeat the process.

Each time, your sequence should get closer and closer to a value which we call the *limit*.**Can you find a relationship between your starting numbers and the limit of the sequence they generate?**

Can you explain why this happens?

Now start with three numbers.

This time, we can generate a sequence where the next number is the mean of the last three numbers.

Check you agree that if we start with $4, 1, 10$, the next number is 5, and the number after that is $\frac{16}{3}$. **What would happen if you continued this process indefinitely?**

Choose some more sets of three starting numbers.**Can you find a relationship between your starting numbers and the limit of the sequence they generate?**

Can you explain why this happens?

Extension

Explore what happens when you have $n$ starting numbers and you generate a sequence where the next number is the mean of the last $n$ numbers.

Try starting with $2, 8$ instead of $8, 2$.

What happens if you keep one of the starting numbers constant and only change the other?

It might be faster to use a spreadsheet or to write a short program to help you identify the limiting value (the value you get if you continue this process indefinitely).

Keep a record of your results.

Share your results.

Test out your ideas.

### 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:

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 support

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

### Possible extension

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.