©Evergreen Public Schools 2011 1 Arithmetic Sequences Recursive Rules Teacher Notes Notes : We will continue work students have done with arithmetic sequences.

©Evergreen Public Schools 2011

1

Arithmetic SequencesRecursive RulesTeacher Notes

Notes: We will continue work students have done with arithmetic sequences.

Vocabulary:arithmetic sequenceexplicit formrecursive form

4/11/2011

©Evergreen Public Schools 2010 2

Practice 7. Look for and make use of structure.

Practice 8. Look for and express regularity in repeated reasoning.

http://www.insidemathematics.org/index.php/standard-7





3

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

LaunchLaunchLaunchLaunchYesterday, we completed the table

and wrote an equation to find the area of L(x) = 2x + 1


5

LaunchLaunchLaunchLaunchWith arithmetic sequence L(x) = 2x + 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


7

Sequences from Unit 1Sequences from Unit 1

Seq Rule Rule

L(x) 3, 5, 7, …

L(x) = 2x + 1

k(x)17, 14, 11, …

k(x) =

We will learn this

today.

The rule in the 2nd column is called the explicit rule.

The rule in the 3rd column is called the recursive rule.


8

How to Write the Recursive Rule

How to Write the Recursive Rule

L(x) = 3, 5, 7, …explicit equation: L(x) = 2x + 1In 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

How to Write a Recursive Rule


a(x) = 7, 9, 11, … explicit rule: a(x) = 2x + 5The 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



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) = 3k(x+1) = a(x) – 3 and a(1) = 7


11

How to Read a Recursive Rule

How to Read a Recursive Rule

For the sequenced(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

Write a Recursive Rule with f(x)

Write a Recursive Rule with f(x)

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

Write rules for each of the sequences.

Write rules for each of the sequences.

Sequence Explicit Rule f(x)

Recursive Rule f(x + 1)

f(x) add 3__, 4, 7, 10, 13,

f(x) = 3x + 1

f(x + 1) = f(x) +3 and f(1) = 4

g(x)8, 14, 20, 26, …

N(x) 34, 30, 26, 22, …


14

Debra’s rulesDebra’s rules

What do you think of Debra’s rules?

Sequence 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) + 6and I(4) = 26

N(x) 34, 30, 26, 22, …

N(x) = N(x-1) – 4 and N(3) = 26


15

Find the rate of change for each sequence.

Find the rate of change for each sequence.

f(x) f(x + 1) Rate of Change

L(x) = L(x-1) + 2and L(1) = 3

L(x+1) = L(x) + 2and L(1) = 3

f(x) = f(x-1) + 3 and f(1) = 4

f(x+1) = f(x) + 3and f(1) = 4

g(x) = g(x-1) + 6and g(1) = 8

g(x+1) = g(x) + 6and 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

Common DifferenceCommon Difference

7, 11, 15, 19, 23The 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, a1 = 24 and the common difference d = 9. What are the first 5 terms of the sequence?


17

53124

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

PracticePractice


19

Placemat Placemat

Write a recursive rule for the sequence p(x)

4, 15, 26, 37, …

Name 1

Name 2

Name 3

Name 4

©Evergreen Public Schools 2011 1 Arithmetic Sequences Recursive Rules Teacher Notes Notes : We will continue work students have done with arithmetic sequences.

Documents