UNIT 2 The general rule
In this unit you will: • describe and justify the general rules for patterns, and • describe patterns in your own words or in algebraic language.
Extending patterns
It is not enough to give only three terms of a number pattern. Three terms do not define a unique number pattern. For example, look at this number pattern of which only three terms are given: 1; 2; 4; … . There are different ways of continuing this number pattern. Here are only two of them: • First way: Multiply each term by 2 to get the next term. The next three terms are 4 × 2 = 8; 8 × 2 = 16; 16 × 2 = 32. So, this pattern could be: 1; 2; 4; 8; 16; 32; … .
• Second way: To get the next term, add 1, then add 2. In other words, add one more each time. So, the next three terms are 4 + 3 = 7; 7 + 4 = 11; 11 + 5 = 16. So, this pattern could also be: 1; 2; 4; 7; 11; 16; … .
We can see that three terms can easily lead to more than one interpretation of a number pattern. We always need more than three terms to be able to fully describe a particular pattern.
The general rule
One way to extend a pattern is to find further terms one by one, but this can be time-consuming. A better way is to find the general rule that governs the pattern. We can then apply this rule to find the general term of the pattern. We call this general term the nth term of the pattern. (Note: n is the term’s position in the pattern.)
• We use the notation Tn to mean the nth term of a pattern. So, T1 is the first term, T2 is the second term, and so on.
• We say that n is the independent variable and Tn is the dependent variable, because the value of Tn depends of the value of n. Once we have found the general rule for a pattern, we can write this rule in words, or in algebraic language.
Unit 2: The general rule
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