Sequences and rules — Mathematics, 11–13
A rule can build a list of numbers or figures, and the best rules jump straight from a term's position to the term.
A rule that makes the list grow
A sequence is a list of terms that follows a rule. The rule can say how to get the next term from the one before: 0.5, 1.5, 4.5, 13.5… (multiply by 3 each time). Or it can give a term from its position: in 3, 6, 9, 12… the term in position n is 3 × n, so position 5 gives 15. To create a sequence, choose a first term and a rule: start at 2 and add 5 each time to get 2, 7, 12, 17… A rule also fills gaps: in 3, 6, ?, 12 the missing term is 9.
Jump straight to the term you want
If all you know is “add 3 to the previous term”, to find the term in position 50 you must do 49 additions in a row. A rule that links position to term gives you the 50th at once. And you can justify it by looking at the picture: in a row of squares made of matchsticks, figure 1 has 4 sticks and each new square adds 3 (top, bottom and right side). That leaves 1 stick on the left of the first square plus n groups of 3: 3 × n + 1.
Figure 20 and the number 100
Rule: figure n has 3 × n + 1 sticks (figures 1, 2, 3: 4, 7, 10). Figure 20? 3 × 20 = 60 and 60 + 1 = 61 sticks, without working out the 19 figures before it. Is 100 a term? Undo the rule: 100 − 1 = 99 and 99 ÷ 3 = 33, a whole number, so yes: it is figure 33. And 50? 50 − 1 = 49 and 49 ÷ 3 is 16 with 1 left over, not a whole number: 50 is not a term of this sequence.
Double the position is not double the term
Figure 2 has 7 sticks. How many will figure 4 have? Many people think “the position doubled, so the term did too” and answer 14. It is a reasonable mistake, because it does work when the rule is just 3 × n, but here the rule is 3 × n + 1 and the “+ 1” does not double: 3 × 4 + 1 = 13. Check with figure 3: 10 + 3 = 13. Before you trust a rule, test it on the terms you already know: 3 × 1 + 1 = 4, 3 × 2 + 1 = 7, 3 × 3 + 1 = 10.
Fences, spreadsheets and programs
Sequences appear whenever something grows step by step in the same way: along a straight fence, n sections need n + 1 posts. In a spreadsheet you write the rule once and drag it down to get as many terms as you like. In Scratch, a loop “repeat 10 times: add 3” builds the 10 terms of a sequence, and changing the rule changes the whole list at once.
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