©Evergreen Public Schools Arithmetic Sequences Recursive Rules Vocabulary : arithmetic sequence explicit form recursive form 4/11/2011

©Evergreen Public Schools Practice Target Practice 7. Look for and make use of structure. Practice 7. Look for and make use of structure. Practice 8. Look for and express regularity in repeated reasoning.Practice 8. Look for and express regularity in repeated reasoning.

©Evergreen Public Schools Learning Target Sequences 3b I can write an arithmetic sequence in recursive form and translate between the explicit and recursive forms. Sequences 2 I can write an equation and find specific terms of an arithmetic sequence in explicit form.

©Evergreen Public Schools LaunchLaunch Yesterday, we completed the table and wrote an equation to find the area of L ( x ) = 2 x + 1

©Evergreen Public Schools LaunchLaunch With arithmetic sequence L ( x ) = 2 x + 1 L (4) = 9. Find the term follows L (4) L (100) = 201. Find the term follows L (100) Find the term follows L (x) Find the term comes before L ( x )

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7 Sequences from Unit 1 SeqRule L ( x ) 3, 5, 7, … L ( x ) = 2 x + 1 k ( x ) 17, 14, 11, … k ( x ) = We will learn this today. The rule in the 2 nd column is called the explicit rule. The rule in the 3 rd column is called the recursive rule.

©Evergreen Public Schools L ( x ) = 3, 5, 7, … explicit equation: L ( x ) = 2 x + 1 In the pattern L ( x ) the next term is 2 more than what I have now. Now is L ( x ) Next is L ( x +1) So rule is L ( x +1) =

©Evergreen Public Schools a ( x ) = 7, 9, 11, … explicit rule: a ( x ) = 2x + 5 The pattern in a is the next is 2 more than what I have now. Now is a ( x ) Next is a ( x +1) So rule is a ( x +1) = a ( x ) + 2 But wait, isn’t this the same rule for L ? L ( x +1) = L ( x ) + 2

©Evergreen Public Schools So the rule needs one more thing. What could that be? We need to know one term in the sequence. L ( x +1) = L ( x ) + 2 and L (1) = 3 k ( x +1) = a ( x ) – 3 and a (1) = 7

©Evergreen Public Schools For the sequence d ( x +1)= d ( x ) – 5 and d (1) = 63 Find the first four terms in the sequence. If d (20) = -33, find d (21) Write the explicit rule

©Evergreen Public Schools What if I wanted to write the rule with L ( x ) or k ( x ) instead of L ( x +1) or k ( x +1) ? L ( x ) = k ( x ) = L ( x ) and k ( x ) are what I have now. What other term do I need? I need what I had before. L ( x – 1) or k ( x – 1)? L ( x – 1) + 2 and L (1) = 3 k ( x – 1) + 2 and a (1) = 5

©Evergreen Public Schools Write rules for each of the sequences. SequenceExplicit Rule f ( x ) Recursive Rule f ( x + 1) f ( x ) add 3 __, 4, 7, 10, 13, f ( x ) = 3 x + 1 f ( x + 1) = f ( x ) +3 and f (1) = 4 g ( x ) 8, 14, 20, 26, … N ( x ) 34, 30, 26, 22, …

©Evergreen Public Schools Debra’s rules What do you think of Debra’s rules? Sequence f(x)f(x) f ( x ) 4, 7, 10, 13, … f(x) = f(x-1) + 3 and f(2) = 7 g ( x ) 8, 14, 20, 26, … g(x) = I(x-1) + 6 and I(4) = 26 N ( x ) 34, 30, 26, 22, … N(x) = N(x-1) – 4 and N(3) = 26

©Evergreen Public Schools Find the rate of change for each sequence. f(x)f(x) f ( x + 1) Rate of Change L(x) = L(x-1) + 2 and L(1) = 3 L(x+1) = L(x) + 2 and L(1) = 3 f(x) = f(x-1) + 3 and f(1) = 4 f(x+1) = f(x) + 3 and f(1) = 4 g(x) = g(x-1) + 6 and g(1) = 8 g(x+1) = g(x) + 6 and g(1) = 8 N(x) = N(x-1) – 4 and N(1) = 34 N(x+1) = N(x) – 4 and N(1) = 34 +2

©Evergreen Public Schools Common Difference 7, 11, 15, 19, 23 The rate of change is called the common difference, d in an arithmetic sequence. Why do you think it is called that? The first term of an arithmetic sequence, a 1 = 24 and the common difference d = 9. What are the first 5 terms of the sequence?

©Evergreen Public Schools Learning Target Did you hit the target? Sequences 3c I can write an arithmetic sequence in recursive form and translate between the explicit and recursive forms. Sequences 2a I can write an equation and find specific terms of an arithmetic sequence in explicit form.

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©Evergreen Public Schools Placemat Write a recursive rule for the sequence p(x) 4, 15, 26, 37, … Name 1 Name 2 Name 3 Name 4