1. Sequences

2. Can you find the pattern?
4, 8, 12, 16, ….
1, -2, -5, -8,…
-2, 0, 1, 3,….
Adding 4
Subtracting 3
None

3. What is a sequence?
An ordered list of numbers with a 1st, 2nd, 3rd, … number

4. Infinite Sequences
A function whose domain is the set of positive integers…
A1, a2, a3,…,an,… are the terms of the sequence

5. Finite Sequence
If the domain is only the first n POSITIVE integers only then the sequence if finite

6. Factorial Notation
If n is a positive integer, n factorial is defined by
n!=1*2*3*4*…*n

7. Arithmetic Sequences
Always look for a common difference between values in a set of numbers
This common value should be added or subtracted from each term to get the next term
Common difference: “d”

8. Recursive Arithmetic Sequences
Recursive Formula
Defines a term in a sequence by relating each term to the term before it
You MUST be given (or find) the previous term to use this formula
Common
Difference
Current
Term
Previous
Term

9. Examples Find the first 5 terms of the sequence:
Is this sequence arithmetic? Write a recursive formula for the sequence.
4, 8, 12, 16,…
yes

10. Explicit Arithmetic Sequences
Explicit Formula
Defines the nth term in terms of n
This allows you to find any term in the sequence without knowing any other terms.
Common
Difference
Current
Term
1st
Term
Term
Number

11. Examples Find the first 5 terms of the sequence.
Is this sequence arithmetic? Write an explicit formula for this sequence.
5, 8, 11, 14, 17,…
yes

12. You Try 
Write a recursive and explicit formula for the following sequence: 3, 7, 11, 15, ….
Find the 16th term in the following sequence:
9, 4, -1, -6,…
Is the formula explicit or recursive and find the first 4 terms.
an= 2*an-1+3 ; a1=3
an= ½*(n-1)

13. Solutions  an= an-1+4 and an= 3+(n-1)*4 an= -5n+14 so a16=-66
Recursive; a1=3 a2=9 a3=21 a4=45
Explicit a1=0 a2=½ a3=1 a4=1.5

14. More Sequences

15. Can you find the pattern?
-1, 1, -1, 1, ….
4, 2, 1, .5,…
1, 3, 5, 7,….
Multiplying by -1
Dividing by 2
Adding 2

16. Geometric Sequences
Always look for a common ratio between values in a set of numbers
This common value should be multiplied or divided by each term to get the next term
Common ratio: “r”

17. Recursive Geometric Sequences
Recursive Formula
Defines a term in a sequence by relating each term to the term before it
You MUST be given (or find) the previous term to use this formula
Common
Ratio
Current
Term
Previous
Term

18. Examples Find the first 5 terms of the sequence:
Is this sequence geometric? Write a recursive formula for the sequence.
16, 8, 4, 2,…
yes

19. Explicit Geometric Sequences
Explicit Formula
Defines the nth term in terms of n
This allows you to find any term in the sequence without knowing any other terms.
Term
Number
Current
Term
1st
Term
Common
Ratio

20. Examples Find the first 5 terms of the sequence.
Is this sequence geometric? Write an explicit formula for this sequence.
4, 8, 16, 32, 64,…
yes

21. You Try 
Write a recursive and explicit formula for the following sequence:
Find the 11th term in the following sequence:
1, -2, 4, -8,…
Is the formula explicit or recursive and find the first 4 terms.
an= 5*an-1 ; a1=-2
an= 5*-3(n-1)

22. Solutions  an= an-1*1/2 and an= 1/2*(1/2)n-1
an= 1*(-2)n-1 so a11=-1024
Recursive; a1=-2 a2=-10 a3=-50 a4=-250
Explicit a1=-5 a2=-15 a3=-45 a4=-135