Lessons · Teaching · patterns, rules and what a function is
A rule that turns any input into its output
A pattern has a rule, and the useful rules say what to do to the position number rather than to the previous term. 'Add 3 each time' cannot tell you the fiftieth term without listing forty-nine of them; 'three times the position, plus one' can.
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What it is for
This is where arithmetic becomes algebra, and it is the idea an elementary examination tests as 'function tables'. A pupil who can only say 'add 3' has a recursive rule; the examination wants the one that jumps straight to any term.
How to think about it
Tabulate position against term. Find the constant difference between terms: that is the multiplier on the position. Then find what must be added or subtracted to fix the first term. Test the rule against a term you did not use to build it.
Worked example
Terms 4, 7, 10, 13 at positions 1, 2, 3, 4: the difference is 3A constant difference means the rule is position times that difference, adjusted.
3 times the position gives 3, 6, 9, 12, which is one short every timeCompare the multiples to the terms and read off the gap.
So the rule is 3n + 1, and the 50th term is 151The whole point: no listing required.
Test on position 4: 3 x 4 + 1 = 13, which matchesTest on a term you did not use to build the rule.
Your turn
Name what the constant difference between terms becomes in the rule.
The constant difference becomes the on the position
Solve one, graded on the server
The trap
Giving 'add 3' as the rule when the question asks for the hundredth term. It is a true description and it does not answer the question.