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.