1. ©Evergreen Public Schools 2011 1 8.1 Arithmetic Sequences Recursive Rules Vocabulary : arithmetic sequence explicit form recursive form 4/11/2011

2. ©Evergreen Public Schools 20102 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.

3. ©Evergreen Public Schools 2011 3 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.

4. ©Evergreen Public Schools 2011 4 LaunchLaunch Yesterday, we completed the table and wrote an equation to find the area of L ( x ) = 2 x + 1

5. ©Evergreen Public Schools 2011 5 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 )

6. ©Evergreen Public Schools 2011 6 ExploreExplore

7. 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.

8. ©Evergreen Public Schools 2011 8 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) =

9. ©Evergreen Public Schools 2011 9 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

10. ©Evergreen Public Schools 2011 10 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

11. ©Evergreen Public Schools 2011 11 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

12. ©Evergreen Public Schools 2011 12 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

13. ©Evergreen Public Schools 2011 13 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, …

14. ©Evergreen Public Schools 2011 14 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

15. ©Evergreen Public Schools 2011 15 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

16. ©Evergreen Public Schools 2011 16 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?

17. ©Evergreen Public Schools 2011 17 5 3 1 2 4 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.

18. ©Evergreen Public Schools 2011 18 Practice

19. ©Evergreen Public Schools 2011 19 Placemat Write a recursive rule for the sequence p(x) 4, 15, 26, 37, … Name 1 Name 2 Name 3 Name 4