Mathematics Developmental Continuum P-10

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Rules For Sequences - More About

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Explanation

A sequence is just a list of numbers (called the terms) that goes on indefinitely. Sometimes if we write the first few numbers of a sequence it can be easy to tell how the sequence keeps going. For example,

10, 20, 30, 40, 50, 60 ... This sequence is easily recognised as being a list of the multiples of 10. Most students will be able to identify 70, 80, 90, 100 as the next four numbers, perhaps without even explicitly identifying to themselves how the pattern is generated.

Of course, it is possible to say that any number, or term, in the above sequence is just 10 more than the term before it. This explains why the next number is 70 (obtained by adding 10 to 60); the number afterwards is 80 (= 70 + 10), and so on.

When terms can be defined using earlier terms of the sequence we say that a recursion rule or a recurrence relation defines the sequence. We do have to say how the sequence starts off, however (eg, we have to say the above sequence starts with 10, otherwise we don't know how to find the second term).

Here is another example of a sequence that can be defined recursively.

1, 2, 4, 8, 16, 32, 64 ... This sequence starts with 1, and the recursion rule is that the next term is double the number before it (so, 2 = 2 × 1, 4 = 2 × 2, 8 = 2 × 4, 16 = 2 × 8, etc). [Note that students may need to be reminded that "doubling" and "multiplying by 2" are equivalent].

In both the examples above, finding out the 50th number in the sequence seems to require us to write out all 50 terms, which might be a bit tedious. With many sequences, including the ones above, we can find a formula that tells us how to find out the values of the terms, using the number of the term we are interested in. This gives a function rule for the sequence. If we have a function rule, then we don't have to write out all those intermediate terms.

Terms of sequence
10
20
30
40
50
60
...
Position or term number
1
2
3
4
5
6
...
Here it is easy to see that the 1st term is 10×1, the 2nd term is 10×2 , the 3rd term is 10×3 , and so on. In general, the nth term will be 10×n, which means that the 50th term will be 10×50.

Some students may be able to see the pattern in terms of specific numbers first, before being able to generalise. To help them make the generalisation get them to predict, and describe how they determined, different terms such as the 40th, the 99th, the 650th, and so on.
Terms of sequence
1
2
4
8
16
32
...
Position or term number
1
2
3
4
5
6
...
Here it is a little harder to determine what is happening. It helps to skip the first couple of terms for the moment, and notice that the 3rd term is 2 × 2 or 22, the 4th term is 2 × 2 × 2 or 23, the 5th term is 2 × 2 × 2 × 2 or 24, and the 6th term is 2 × 2 × 2 × 2 × 2 or 25.

This suggests that the 50th term is 249 (that is 562949953421312 if you want it worked out!).

More generally, the nth term is 2n-1. This formula works for the first two terms as well: the formula says the 2nd term is 22-1 which is 21 or 2; and the 1 st term is 21-1 which is 20 = 1. This matches the sequence.

And, of course, if someone tells us either a recursion rule or a function rule, then we can work out the corresponding sequence:

Recursion rule:

Start with two numbers: 1, 1

Recursion rule: New number is the previous two added together

 This recursion rule gives the famous Fibonacci numbers.
Position or term number
1
2
3
4
5
6
7
8
...
Applying the rule
1+1
1+2
2+3
3+5
5+8
8+13
Terms of sequence
1
1
2
3
5
8
13
21
...

Note how there is no "applying the rule" for the first two terms. We are TOLD what these are; they allow us to get the sequence started.
Function rule:

The nth term is given by n2 + n.
Position or term number
1
2
3
4
5
6
7
8
...
Applying the rule
12+1
22+2
32+3
42+4
52+5
62+6
72+7
82+8
Terms of sequence
2
6
12
20
30
42
56
72
...

Note that if we were only given the sequence of numbers 2, 6, 12, 20, 30, 42, 56, 72 ... it would be quite difficult to come up with a recursion rule, even though it might be possible to see that the successive differences between the numbers follow a pattern (4, 6, 8, 10, 12, 14 ...).

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Summary of examples

Some more examples of sequences are given in the table below, along with the recursion rules and function rules that produce them.

The table shows:

  • that a recursion rule gives the next term by calculations on the previous term (or terms, as in sequence (e) below),
  • that a function rule gives each term by calculation on its position number (first, second, etc),
  • that recursion rules are often simpler than function rules (but certainly not always),
  • that not all sequences have simple recursion rules or function rules, but these are not dealt with at this level.
  Sequence of Numbers

Recursion rule:

To get the next number:

Function rule:

To get the number in a certain position:

(a) 1, 2, 3, 4, 5 ... add 1 to the current number it is equal to its position number
(b) 2, 4, 6, 8, 10 ... add 2 to the current number double the position number (eg 2 = double 1, 4 = double 2 ...)
(c) 10, 30, 50, 70, 90, 110 ... add 20 to the current number multiply the position number by 20 and subtract 10
(eg 30 = 2 × 20 - 10, 50 = 3 × 20 - 10)
(d) 1, 4, 9, 16, 25 ... add the next odd number square the position number
(e) 1, 1, 2, 3, 5, 8, 13, 21 ... (Fibonacci numbers) add the previous two terms together very complicated