Number sequences

"J'ai seul la clef de cette parade sauvage"

Rimbaud, 'Les Illuminations'

The topic is sometimes referred to as 'number patterns' in conversation and in the National Curriculum but the word 'pattern' is heavily overworked - it carries a connotation of repetition but this may range from a vague resemblance (a pattern of events or weather) through varying degrees of exactness (tigers' stripes, dress-making, foundry pattern etc) and the more specific mathematical term 'sequence' is preferable.

The Curriculum implies, and test papers sometimes demand, that pupils should be able to continue a sequence represented by a few terms. This is impossible since a sequence is only completely defined when a rule for generating its terms is specified. It is a fundamental theorem that a 'rule' which fits the given terms and any arbitrary continuation can be found. The rule might be specified indirectly, by identifying the sequence as 'linear' or 'quadratic' for example (but never is) . The Curriculum nominally restricts the requirement to these two cases but in fact other types have been found in test papers.

2. Definitions		6. Other sequences
3. Guesswork		8. Health warning
4. Linear sequences		9. Trick questions
5. Quadratic sequences

1. Definitions

The most important thing about a number sequence is the rule which allows you to calculate its terms. A simple example is 2,4,6,8,.....

The '.....' is a mathematical way of saying 'and so on' and an obvious guess for the next number is 10 but this is only a guess. It is not the only possible way of continuing - in fact you can put any numbers you like next and still find a rule which duplicates the whole sequence. If 'even numbers' is written at the end of the line, this effectively gives the rule for calculating the following terms since it is assumed that everyone knows what the even numbers are. You can't be certain how the sequence continues until the rule is given, and then it is not actually necessary to write down the first terms of the sequence.

2. To talk mathematically about sequences we need a convenient short way of identifying particular terms and we write T_n to mean the n'th term of the sequence eg T₃ for the sequence above is 6, T₄ is 8 etc. T_n+1 means the next term after T_n.

To specify a sequence we must have a rule for calculating T_n for any value of n and there are two ways of doing this - either a rule for calculating T_n directly from n or one for calculating the next term from the last eg for the 'squares' sequence1,4,9,16,.... we can use either

T_n = n × n or T_n+1 = T_n + 2n +1

In this case both rules are simple but sometimes one is much more complicated than the other eg for the Fibonacci sequence 1,1,2,3,5,8,.... the rule for the next term is simple T_n+1 = T_n + T_n-1 - but a direct rule for the n'th term is complicated and difficult to find.

3. Guessing an answer

Although the continuation of a sequence can never be known with certainty unless the rule generating it is known it is quite often possible to make a sensible guess at a rule that might apply by looking at a few terms. The rule must fit all the given terms exactly, of course, and the more terms available the more likely it is to be right. Such guesses can serve a useful practical purpose as clues in investigating the situation producing the sequence and perhaps finding proof that the rule is the correct one.

A useful trick is to look at the diferences between terms because these often form a simpler sequence. There are a number of typical cases which may help to find possible rules.

4. Linear sequences

The first type, where the differences are constant, is quite common eg

N	1		2		3		4		5
sequence:	5		7		9		11		13
difference:		2		2		2		2

If we assume the differences remain constant the following values can be filled in easily by extending the table and we can see that a possible rule is T_n+1 = T_n + 2.

With this type of series another useful trick is to work backwards from the start to find the term for N = 0 ie in this case 3. Since we are adding 2 to this every time N increases by one we get a possible 'term' rule T_n = 3 (the value for N = 0) + 2N

N	(0)		1		2		3		4		5		6		7
sequence:	(3)		5		7		9		11		13		(15)		(17)
difference:		(2)		2		2		2		2		(2)		(2)

Note: The bracketed numbers are the new values given by assuming the differences remain constant.

A similar example is

N	(0)		1		2		3		4		5		6		7		8
Sequence	(23)		20		17		14		11		8		(5)		(2)		(-1)
differences		(-3)		-3		-3		-3		-3		(-3)		(-3)		(-3)

giving a possible 'increment' rule T_n+1 = T_n - 3 and a corresponding 'term' rule T_n = 23 - 3N

5. Quadratic sequences

A slightly more difficult example is

N	1		2		3		4		5
Sequence	2		6		12		20		30
difference		4		6		8		10

The sequence can be extended by noticing that the differences increase by 2 each time, but finding a possible rule is not so easy

N	1		2		3		4		5		6		7
Sequence	2		6		12		20		30		(42)		(56)
difference		4		6		8		10		(12)		(14)

The 'difference' sequence is similar to the main sequence in the first two examples and if we put in the difference for N = 0, we can find a possible rule for the differences :-

N	(0)		1		2		3		4		5
Sequence	(0)		2		6		12		20		30
difference		(2)		4		6		8		10
2nd difference				(2)		2		2		2

ie N'th difference = 2N + 2 and hence a possible 'increment' rule for the original sequence isT_n+1 = T_n + 2N + 2

Finding the 'term' rule is more difficult and needs some algebra not yet covered but noticing that this is is very similar to the 'squares' sequence a possibility is T_n+1 = T_n + (2N + 1) + 1

A previous example shows that T_n+1 = T_n + 2N + 1 gives the 'squares' sequence T_n = N×N and since an extra 1 is added every time N increases guess the rule might be T_n = N×N + N

Checking against the terms given, this fits perfectly and in fact it is the correct answer in this case.

6. Other sequences

The sequences specified in the Curriculum are the linear and quadratic types discussed above but other types have been noted in test papers from time to time. The Fibonacci sequence looked at previously is of a different type :-

N	1		2		3		4		5		6		7		8
Sequence	1		1		2		3		5		8		(13)		(21)
difference		0		1		1		2		3		(5)		(8)

The differences here are not constant but we notice that the difference is the same as the main sequence itself moved along one place, so that we can extend the sequence by copying the original values into the differences, and a possible 'increment' rule is T_n+2 = T_n+1 + T_n

The 'term' rule is too difficult for most people to work out in this case. A second example of this type is the sequence

N	1		2		3		4		5		6		7
Sequence	3		9		27		81		243		(729)		(2187)
difference		6		18		54		162		(486)		(1458)

Here the differences are just twice the terms and the sequence may be continued as shown by the bracketed terms. The differences have not really helped much however since it is just as easy to spot that each term is 3 times the previous one.ie T_n+1 = 3 × T_n

The differences are more helpful in the following example

N	1		2		3		4		5		6
Sequence	2		7		17		37		(77)		(157)
difference		5		10		20		(40)		(80)

The differences are doubled each time so the sequence might be extended by assuming that this relation continues, giving the values in brackets. Finding a rule depends on noticing that the differences are just 3 larger than the terms, so we can say T_n+1 = T_n + (T_n +3) = 2T_n + 3

7. In some cases the difference table technique is no help at all eg

N	1		2		3		4		5		6		7		8		9		10
Sequence	1		2		3		5		7		11		13		17		19		23
difference		1		1		2		2		4		2		4		2		4

The first of these is the sequence of prime numbers - there is no rule for generating these - while the second is a random generator which, although it has a definite rule (and is therefore predictable in principle), is specifically designed to make successive numbers appear unrelated, and hence unpredictable in practice.

8. Mental Health Warning

The following sequences all have well defined rules which are quite similar but as you see their sequences agree for several terms and then behave quite differently, demonstrating the fact that no method of guessing the continuation from a limited number of terms can be absolutely certain. Use of the difference technique is often helpful in finding a credible continuation and sometimes in finding possible rules to generate the sequence. In some cases it will not help at all and in others may give the wrong answer altogether, so BEWARE!!

N	1		2		3		4		5		6		7		8		9
Sequence A	1		3		9		27		81		243		729		2187		6561
Sequence B	1		3		9		27		81		115		89		11		33

A similar set all of which have definite and quite similar rules are

N	1		2		3		4		5		6		7		8		9
Sequence A	1		4		13		40		121		364		1093		3280		9841
Sequence B	1		4		13		40		121		108		69		208		113
Sequence C	1		4		13		40		121		109		73		220		151

9. Trick questions

Test papers sometimes include trick questions like, for example,

Continue the following sequences

(a)	S	M	T	W	T
(b)	R	O	Y	G	B
(c)	1	32	60	91	121	152 ....
(d)	Y	G	B	B

The answers are
(a) F S - Friday Saturday
(b) I V - Indigo Violet - conventional colours of the rainbow
(c) 182, 213, 244 ... first day of July, August, and September..
(d) P B - Pink Black - colour of snooker balls in order of value

These all seem rather trivial when the answer is revealed and in my opinion are not legitimate mathematical questions. They violate a fundamental principle of mathematics - that the context and nature of the objects involved must be made clear. Up to this point the title 'Number Sequences' makes it clear that we are talking about sequences of numbers, although to be strictly accurate we might have said 'Integer Sequences' since we have not dealt with any other type of number.

If it were said at the outset that (a) relates to abbreviatons for English day-names, (b) to abbreviations for colours, and (c) to Julian dates (ie number of day counting from Jan 1st), and (d) is about snooker balls the problems are seen to be devoid of any mathematical or logical content.