### Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

### Zooming in on the Squares

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?

# Litov's Mean Value Theorem

##### Stage: 3 Challenge Level:

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?

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.