Sequences

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

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

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

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

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

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”

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

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

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

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

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)

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

More Sequences

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

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”

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

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

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

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

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)

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