Download Notes for 2B: Finding Sequence Rules

Notes for 2B: Finding Sequence Rules
Recall: What is a sequence rule
A sequence rule is a rule that tells us how to find the next number in a number patter (a sequence).
For example, Start at 3, go up by 10 is a sequence rule which generates the following sequence
3, 13, 23, 33, 43, …
Finding a Sequence Rule from a Pattern
What rule do you think generates this sequence: 5, 7, 9, 11, 13, 15, … ?
Answer: Start at 5, go up by 2.
So how do you find a sequence rule?
Addition Rules
Well, the best strategy is to start by seeing if the numbers go up by the same amount each time.
If they do, it is easy!!!
For example:
18, 25, 32, 39, 46, 53, 60, 67, …
Check how much we are going up by each time. You can write these increases above or below the
main sequence, in the space between 2 numbers. If you do it for this sequence you get
So the rule is: Start at 18 (that should have been easy to find) and then add 7 each time.
Subtraction Rules
Not all sequences have subsequent numbers getting bigger and bigger.
For example, 20, 17, 14, 11, 8, 5, 2, -1, -4, …
This sequence is going down. The difference between any 2 consecutive numbers is 3.
So the rule would be: Start at 20, go down by 3 each time.
Multiplying Rules
If you don't get a constant number, the sequence rule could be quite complex.
In Year 7, you are really only expected to be able to notice patterns that go up by a constant
amount each time, OR, if it doesn't, check to see if it is multiplying or dividing each time.
If the difference between 2 consecutive numbers is not constant, then it might be a multiplying
rule. To check to see if this works, see how many times the first number goes into the second,
Then see how many times the second number goes into the third. Then how many times the third
goes into the fourth. This may reveal a PATTERN.
Example:
5, 15, 45, 135, …
Solution:
Adding or subtracting won't work because it goes up by different amounts each time.
Try multiplying:
Hmm, notice the pattern. It gets 3 times bigger each time.
So the rule is: Start at 5, and multiply by 3 each time.
Dividing Rules
Look at the following sequence:
800, 400, 200, 100, 50, 25, 12.5, …
What is the rule?
Solution:
It's going down so it can't be addition or multiplication.
It goes down by different amounts each time so it's not subtraction.
Try division.
400 goes into 200 2 times, so its divide by 2.
200 goes into 400 2 times, so its divide by 2.
100 goes into 100 2 times, so its divide by 2.
There's a pattern!!!
So the rule is: Start at 800, and divide by 2 each time.
Continuing a Sequence
Once you know what the sequence rule is, you can keep generating more terms in a sequence.
For example;
Find the next 3 terms in this pattern: 4, 10, 16, 22, 28, 34, 40, 46, …
Solution: Notice the rule is Start at 4. Go up by 6 each time. So by following the “go up by
6 each time” rule, the next 3 terms must be 52, 58, 64.
Filling in Blanks
Note, if I have just 2 consecutive numbers, then I can come up with 2 possible rules:
e.g. For 4, 12 it could be up by 8, or multiply by 3.
And for 20, 10, it could be down by 10, or divide by 2.
So if you have a filling in the blanks question, find the possible rules and test them by continuting
the pattern and seeing they get to the other numbers.
Example:
3, 5, __, __, 11
I think the rule might be up by 2 from the first 2 terms. By continuing this rule, I get 3, 5, 7, 9, 11.
Notice I finished on 11 just as I was supposed to. So I can confident the rule is correct.
Another Example:
2, 4, __, __, 32.
I might start by thinking up by 2, but completing it gives me 2, 4, 6, 8, 10. I finished on 10 but I
was supposed by finish on 32. So that rule is wrong. Try the multiply by 2 rule instead. Then I get
2, 4, 8, 16, 32. This is exactly what I needed. So I know the rule is multiply by 2 and that the
missing numbers must be 8 and 16.
Working Backwards
The sequence rule tells us how to go FORWARD and create new terms.
For example 5, 8, 11, 14, 17, 20, 23, … etc.
The rule is +3 each time. Now, the backwards rule, the rule for working backwards would be to
subtract 3 each time.
Backwards rules are useful for filling in gaps at the start.
Example:
Complete the gaps for __, __, 7, 10, 13...
Solution:
Using the 3 numbers I have, the rule seems to be “go up by 3”. So the backwards
rule would be to go down by 3.Continuing on backwards from 7, I find that the
other terms are 4, and 1. So the sequence is 1, 4, 7, 10, 13...