Start with two numbers, say 8 and
2. This is the start of a sequence of numbers.
The rule is that the next number in
the sequence is the average of the last two numbers.
So the
next number is 1/2 of (8 + 2), which equals 5,
so the sequence becomes 8, 2, 5
The next number is 1/2 of (2 + 5), which equals 7/2,
so the sequence becomes 8, 2, 5, 7/2
Continue the sequence until you know what will happen when you
continue this process indefinitely.
Choose two different starting numbers and repeat the
process.
Continue exploring with different start numbers, trying to
discover what the rule is for finding the limiting value.
Can you find a relationship between the two start numbers and the
limiting value?
Can you explain why it works?
Now start with three numbers, say
4, 1, 10.
The new rule is that the next
number is the mean of the last three numbers.
So the next
number is 1/3 of (4 + 1 + 10), which equals 5,
so the sequence becomes 4, 1, 10, 5
The next number is 1/3 of (1 + 10 + 5), which equals 16/3,
so the sequence becomes 4, 1, 10, 5, 16/3
Continue this sequence and find the limiting value as the process
is continued indefinitely.
Can you find a rule for finding the limiting value in this
case?
Can you find a relationship between the three start numbers and
the limiting value?
Can you explain why it works?
General rule
Now explore what
happens when you have
start numbers and the rule for working out the next number
changes to finding the average of the last
numbers.