How to Succeed with Maths

When numbers are put together into a sequence there is a connection between a given number, the one before it and the one after it.

Here is the simplest number sequence:

1  2  3  4  5  6....etc.

If you are asked to describe the pattern of this sequence you can say "Add 1 to any given number to get the next number".

Here is another sequence:

2  4  6  8  10....etc.

The pattern here is "Add 2 to any given number to get the next number". You can express patterns in your own words provided they are clear e.g. "it goes up in 2's" "add 2 every time" etc.

Each number in a sequence can be numbered to show its position in the sequence. In the example above the first term (as it is called) is 2, the second term is 4, the third term is 6 etc. These can be shown in a table:
Term (n) 1 2 3 4 5
Sequence (s) 2 4 6 8 10


It is possible for us to create a rule which can be used to give us any number in the sequence when we know the term. For example, What do you think the 500th term of this sequence is?

Well, you could write out the numbers in the sequence from 2 upwards 500 times or you could use a bit of maths to make things more bearable.

There are 3 simple steps you can use to find a rule:

1) Find out the pattern - in this case it's +2 (it could be +3 or -4 etc.)
2) If the pattern has +2 in it the rule will have x2 in it (if a negative number, the rule will have divide by that number in it)
3) Describe the rule in words - "To get the number in the sequence times the term by 2"
4) Describe the rule as a formula i.e.    s = 2n

So, the 500th term is 1,000.

Consider another example:
Term (n) 1 2 3 4 5
Sequence (s) 3 5 7 9 11

Pattern:
"Add on 2 every time"

Rule:
1) Pattern is +2 again
2) So the rule will have x2 in it
3) "Multiply the term by 2 and add 1 to get the number in the sequence"
4) s = 2n + 1

The only other type of sequences you'll need to worry about are those which end up with a rule which has n2 in it.

Consider the example:

Term (n) 1 2 3 4 5
Sequence (s) 2 8 18 32 50

Here we add 6 to 2, then 10 to 8, then 14 to 18 etc. Oh no, the difference isn't the same - how do we deal with this?

First work out the differences and write them in their own sequence:

6   10    14    18

Now do the same again, which gives the sequence:

4  4  4

Here it took us 2 steps before the differences were the same (i.e 4) this means we need n2 in the rule.

If it takes 3 steps we need n3 in the rule etc.

Now we need to use a bit of thought to work out the rule given that we now know that it has n2 in it somewhere.

If s = n2 we end up with the sequence 1   4   9   16   etc.

Now you need to spot that this is half of what we need for our sequence (i.e. we start with 2 instead of 1, then have 8 instead of 4 etc.)

So, we can finally say that the rule is:    s = 2n2.