©Evergreen Public Schools Learning Target Target 12 Level 3 I can write an arithmetic sequence in recursive form and translate between the explicit and recursive forms. Target 12 Level 2 I can write an equation and find specific terms of an arithmetic sequence in explicit form. (Target 7, Level 3)

©Evergreen Public Schools LaunchLaunch 1. Complete the table and write an equation to find the area of L ( x ).

©Evergreen Public Schools LaunchLaunch 2.With arithmetic sequence f ( x ), What term follows f (4)? What term follows f (100)? What term follows f (x)? What term comes before f (x)?

©Evergreen Public Schools ExploreExplore

5 Sequences from Unit 3 Read at L ( x ) and a ( x ). Write a rule for both boxes on the bottom of the pages. SeqRule (on left)Rule (on right) L(x)L(x) a(x)a(x)

©Evergreen Public Schools Sequences from Unit 3 SeqRule L(x)L(x) a(x)a(x) We will talk about this tomorrow. The rule in the 2 nd column is called the recursive rule. The rule in the 3 rd column is called the explicit rule.

©Evergreen Public Schools How to Write the Recursive Rule The pattern in L ( x ) is the next is 2 more than what I have now. Now is L ( x ) Next is L ( x +1) So rule is L ( x +1) = L ( x ) + 2

©Evergreen Public Schools How to Write a Recursive Rule 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 How to Write a Recursive Rule 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 a ( x +1) = a ( x ) + 2 and a (1) = 5

©Evergreen Public Schools How to Read a Recursive Rule For the sequence d ( x +1)= d ( x ) – 5 and d (1) = 63 Find the first four terms in the sequence. If d (20) = -37, find d (21)

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

©Evergreen Public Schools Write rules for each of the sequences. SequenceRecursive Rule f ( x ) Recursive Rule f ( x + 1) f ( x ) 4, 7, 10, 13, … I ( 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 I ( x ) 8, 14, 20, 26, … I(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 I(x) = I(x-1) + 6 and I(1) = 8 I(x+1) = I(x) + 6 and I(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? Target 12 Level 3 I can write an arithmetic sequence in recursive form and translate between the explicit and recursive forms. Target 12 Level 2 I can write an equation and find specific terms of an arithmetic sequence in explicit form. (Target 7, Level 3)

©Evergreen Public Schools Practice Arithmetic Sequences KUTA Problems #23 – 30.

©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