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Patterns and Sequences Henrico County Public School Mathematics Teachers Patterns and Sequences • Patterns refer to usual types of procedures or rules that can be followed. • Patterns are useful to predict what came before or what might come after a set a numbers that are arranged in a particular order. • This arrangement of numbers is called a sequence. For example: 3,6,9,12 and 15 are numbers that form a pattern called a sequence. • The numbers that are in the sequence are called terms. Patterns and Sequences Arithmetic sequence (arithmetic progression) – a sequence of numbers in which the difference between any two consecutive numbers or expressions is the same Geometric sequence – a sequence of numbers in which each term is formed by multiplying the previous term by the same number or expression Arithmetic Sequence 1 Find the next three numbers or terms in each pattern. 7, 12, 17, 22,... 5 5 5 Look for a pattern: usually a procedure or rule that uses the same number or expression each time to find the next term. The pattern is to add 5 to each term. The Next Three Numbers 7, 12, 17, 22... Add five to the last term The next three terms are: 27, 32, 37. 22 5 27 27 5 32 325 37 Arithmetic Sequence 2 Find the next three numbers or terms in each pattern. 45, 42, 39, 36... 45, 42, 39, 36... (3) (3) (3) Look for a pattern: usually a procedure or rule that uses the same number or expression each time to find the next term. The pattern is to add the integer (-3) to each term. The Next Three Numbers 2 45, 42, 39, 36... Add the integer (-3) to each term The next three terms are: 33, 30, 27. 36 (3) 33 33 (3) 30 30 (3) 27 Geometric Sequence 1 Find the next three numbers or terms in each pattern. 3, 9, 27, 81... 3 3 3 Look for a pattern: usually a procedure or rule that uses the same number or expression each time to find the next term. The pattern is to multiply each term by three. The Next Three 1 3, 9, 27, 81... Multiply each term by three The next three terms are: 243, 81 3 243 243 3 729 729, 2187 729 3 2187 Geometric Sequence 2 Find the next three numbers or terms in each pattern. 528, 256, 128, 64... 528, 256, 128, 64... 2 or 1 2 2 or 1 2 2 or 1 2 Look for a pattern: usually a procedure or rule that uses the same number or expression each time to find the next term. The pattern is to divide each term by two. The Next Three 2 528, 256, 128, 64... Divide each term by two The next three terms are: 64 2 32 or 64 1 64 32 1 2 2 32, 16, 8. 32 2 16 or 32 1 32 16 1 2 2 16 2 8 or 16 1 16 8 1 2 2 Note To divide by a number is the same as multiplying by its reciprocal. The pattern for a geometric sequence is represented as a multiplication pattern. For example: to divide by 2 is represented as the pattern multiply by a half. 16 2 8 is the same as 16 1 16 8 1 2 2 Patterns & Sequences Decide the pattern for each and find the next three numbers. a) 7, 12, 17, 22, … a) 27, 32, 37 b) 1, 4, 7, 10, … b) 13, 16, 19 c) 2, 6, 18, 54, ... c) 162, 486, 1548 d) 20, 18, 16, 14, … d) 12, 10, 8 e) 64, 32, 16, ... e) 8, 4, 2