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Number Patterns . . . .with pictures! Look at this pattern of counters How many counters will there be in the next pattern? Here is a table to show the number of Counters in each shape Position of shape Number of counters 1 3 2 3 4 6 9 12 How many counters are added each time? The term-to-term rule is + 3 We can also find a position-to-term rule Shape position Number of counters 1 2 3 4 3 6 9 12 In words: number of counters equals 3 × position number Using algebra: c=3×n c = 3n Can you predict how many counters there Would be in shape number 20? Generating Sequences 1. We can do this by using a term-to-term rule Generate the first five terms of these sequences and then describe them in words. Ist term 10 Term-to-term rule Add 3 100 Subtract 5 2 double Generating Sequences 2. We can also do this by using a position-to-term rule Write the first five terms of each of these sequences where the n th term is: a) n + 3 c) 2n – 0.5 b) 105 – 5n d) 4n What is the 20th term for each sequence? Special Number Sequences Square Numbers 1×1 2 ×2 3 ×3 4 ×4 1, 4, 9, 16 25, 36, 49 … These are the first seven square numbers Special Number Sequences Cube numbers 1 ×1 ×1 2 ×2 ×2 3 ×3 ×3 1, 8, 27, 64, 125, 216 … 4 ×4 ×4 Special Number Sequences Triangle numbers 1, 3, 6, 10 15, 21, 28, 36 … Finding a rule for the n th term Remember that: A number sequence is a set of numbers in a given order Each number in the sequence is called a term Look at this number sequence: 5, 8, 11, 14, 17 … We want to find a rule to find the n th term this will enable us to work out any number in the sequence, for example, the 50th term, or any term start by giving each term a position number… 1 2 3 4 5 ….. n 5, 8, 11, 14, 17 What does the sequence go up by? That’s right, +3 So what’s the 50th term then? 152 because when n is 50, 3 x 50 + 2 = 152 1 2 3 6 3 4 5 ….. n 12 15 …. 5, 8, 11, 14, 17 9 3n Because the sequence is adding 3 from one term to the next, our rule must involve the 3 times table and to complete the rule, we need to + 2 The complete rule is 3n + 2