Number Sequences

 

If the numbers get bigger ‘slowly’ the rule is likely to be an adding one:

 

eg.        5, 9, 13, 17, 21…

            (rule +4)

 

If the numbers get bigger ‘quickly’ the rule is likely to be a multiplying one:

 

eg.        5, 15, 45, 135…

            (rule x3)

 

Differences

 

Sometimes writing differences between the numbers helps to ‘see’ the pattern

 

eg.       

	1		3		8		18		35
1st diff’s		2		5		10		17
2nd diff’s			3		5		7
3rd diff’s				2		2

 

Now, by extending this pattern, the next numbers can be found.

 

1		3		8		18		35		61		98
	2		5		10		17		26		37
		3		5		7		9		11
			2		2		2		2

 

A sequence is a list of numbers that follow a pattern or rule.  Each number is called a “term”

 

eg. 3, 7, 11, 15, 19….etc…..

 

1^st term (t₁) = 3

2^nd term (t₂) = 7

3^rd term (t₃) = 11

 

This can be shown in a table

 

	t1	t2	t3	t4	t5
	3	7	11	15	19	etc
OR use ‘n’ to stand for the term number
n	1	2	3	4	5
sequence	3	7	11	15	19

 

ie. In the 3^rd term

t₃ = 11 and n = 3

 

In the 5^th term

t₅ = 19 and n = 5     etc.

 

To help find the rule, or pattern, work out the differences between the numbers in the sequence

 

eg.

n	1		2		3		4		5
seq	3		7		11		15		19
diff		+4		+4		+4		+4

 

 

ie.      3 + 4 = 7

          7 + 4 = 11

          11 + 4 = 15 etc

OR

          t₁ + 4 = t₂

                   t₂ + 4 = t₃

          t₃ + 4 = t₄ etc

 

If the differences are all the same, then the rule will be based on that multiplication table.  In this case, all the differences are 4 so the rule will be based on the 4 times table, so test it

 

	Sequence
	3
	7
	11
	15

 

In each case, the answers are 1 more than the numbers in the sequence, so adjust your rule.  Try  

 

	Sequence
	3
	7
	11
	15

This is now correct

 

The rule is  

 

eg.2   7, 10, 13, 16, 19….etc….

 

n	1		2		3		4		5
seq	7		10		13		16		19
diff		+3		+3		+3		+3

ie.  

 

	Sequence
	ie. Add 4 in each     case   7
	10
	13
	16

 

So the rule is