2.1 Term-to-term and position-to-term rules
For three of the sequences in Figure 6 you probably thought about how to get from one term to the next, and continued that pattern. You could have done this by adding 5, doubling, or drawing another triangle. These are all ways of describing the change between any two consecutive terms, so this is called finding a term-to-term rule.
It would be much more difficult to identify a term-to-term rule for the other two sequences in Figure 6. Instead, you need position-to-term rules. This type of rule connects the position number (1, 2, 3, etc.) to the associated term. A table is a useful representation to show the relationship between position number and the term.
Position | 1 one | 2 two | 3 three | 4 four | 5 five | 6 six | n |
Term (letter sequence) | O | T | T | F | F | S | ? |
Term (fraction sequence) | 1 | 1/4 | 1/9 | 1/16 | 1/25 | 1/36 | 1/n^{2} |
For the letter sequence, the position-to-term rule is ‘the term is the first letter of the position number as an English word’.
For the fraction sequence, ‘the term is the fraction 1 divided by the position squared’. You can write this symbolically as
Term-to-term rules describe the pattern you see by moving along a row of the table.
Position-to-term rules describe the pattern you see when moving down from the position row.
The rule must describe a general pattern that holds for every term of the sequence.
Learners often prefer looking for term-to-term rules. This is for two reasons. First, they only need to look along the changes in one variable – along the terms of the sequence. Second, term-to-term rules involve simpler operations. However, a term-to-term rule is not helpful if you want to find a term that is far away from the ones that are known.
The position-to-term rule is more powerful.
Activity 4 Position-to-term rules
Imagine finding the 37th term in each of the five sequences in Figure 6.
Which one would be easiest?
Which one would be hardest?
Write the position-to-term rules for the first two sequences.
Discussion
Sequence | 37th term | Position-to-term rule |
5, 10, 15, 20, 25 | 185 | Multiply position number by 5. n^{th} term is 5n |
3, 6, 12, 24, 48 | 3 × 2^{36} = 206 158 430 208; most calculators will not display all of these digits. | Raise 2 to the power of one less than the position number. Multiply the answer by 3. n^{th} term is 3 x 2^{n-1} |
1, 1/4, 1/9, 1/16, 1/25, ... | 1/ 37^{2} | n^{th} term is the fraction 1 divided by the position squared. n^{th} term is1/n^{2} |
O, T, T, F, F | T | take the first letter of the position number |
There is another reason to emphasise position-to-term rules. Sequences are an early example of functions. If we relate this to the function machine: the position number is the input, the term is the output and there is a rule that connects them.
When learners start to work with sequences, they work with specific values of the position numbers: 1, 2, 3, etc. These are not indeterminate quantities, so they are thinking numerically. The algebraic thinking only starts when learners consider what the rule is for a general input number, which is usually denoted by the symbol n. The corresponding term is called the nth term.
It is relatively easy to understand how the position number can change, since it follows the natural numbers. So, although, teachers may never mention the word ‘function’ when teaching sequences, by emphasising position-to-term rules they are nevertheless preparing the underlying idea of a function as a relationship between two variable quantities.