3. When rolling 2 dice, what is the probability of rolling a a number that is divisible by 2 ? a a number that is divisible by 2 ? Quiz Michael Jordan is a 92% free throw shooter. If he takes 20 free throws, what is the probability of him making 15 baskets? (Hint: which term in the binomial expansion above will help you calculate the answer?) expansion above will help you calculate the answer?) 2.When rolling 2 dice, what is the probability of rolling a sum of 6? a sum of 6?

HOMEWORK Section 9-4 Section 9-4 (page 728) (evens) 2-22, 26, 30, 32, 40, 46 (16 problems)

9-4 Sequences

What you’ll learn about  Infinite Sequences  Limits of Infinite Sequences  Arithmetic and Geometric Sequences  Sequences and Graphing Calculators … and why Infinite sequences, especially those with finite limits, are involved in some key concepts of calculus.

What is a sequence? 5, 10, 15, 20, 25 2, 4, 8, 16, 32, …2ⁿ,… “…” means the numbers continue (forever) unless there is an ending number as indicated by: “...20” there is an ending number as indicated by: “...20” Finite sequence Ininite sequence Defines a “rule” so that you can find the “kth” term  also an ininite sequence the “kth” term  also an ininite sequence Sequence: an ordered progression of numbers (a list) Common Error: A “series” is something total different than a “sequence. different than a “sequence.

Explicitly Defined Sequences Where (and k = 1,2,3,…) Your Turn: 1. Find the first 4 terms of the following explicitly defined sequence. Also find the 100 th term. sequence. Also find the 100 th term. In explicitly defined sequences you can calculate the “kth” term directly  you don’t have to find any term before it term directly  you don’t have to find any term before it in order to find the “kth” term. in order to find the “kth” term.

Recursively Defined Sequences Gives a way to show the relation between elements of the sequence. between elements of the sequence. For all n > 1 Specifies the initial element of the initial element of the sequence. sequence. Find the 3 rd element of the series: of the series: Often times you must calculate all of the previous elements in order to determine any particular element. 4, 6, 8, 10, 12,…

Your Turn: 2. Find the 4 term of the following recursively defined sequence. sequence. 3. What is the 100 th term? For: n ≥ th term = 1 st term – 99*4

Remember “end behavior” of a function? We are also concerned with the “end behavior” of sequences. What number is this sequence getting closer and closer to? The sequence approaches (but never reaches) zero. We can say that the sequence converges (to some number). What do you think we call the number a sequence converges to? The limit of the sequence.

What about this sequence? What is the limit of this sequence? It has no limit because it doesn’t converge to any number. This sequence therefore diverges. It is a divergent sequence. 1, 2, 4, 8, 16, 32, 64, … Where (and k = 1,2,3,…) Your Turn: 4. Does this sequence converge? If so what is the limit? 5. Does this sequence converge? If so what is the limit?

Limit of a Sequence

Finding Limits of Sequences Converge or Diverge? To use the definition of the limit of a sequence, we need to come up with an formula to find its members. we need to come up with an formula to find its members. For: n ≥ 1 Can you find a formula for this sequence? What happens to the sequence as “n” approaches infinity?

More of a challenge: Find the limit of the sequence if it exists. “factor out” the “3n” term (the smallest power of ‘n’ that occurs both the numerator and denominator. both the numerator and denominator. Now simplify Not so much of a challenge after all. challenge after all. If I can get all the “n’s” into the denominators, I know what happens to a fraction when the denominator gets infinitely big.

More of a challenge: Find the limit of the sequence if it exists. Not so much of a challenge after all. challenge after all.

Your turn: Find the limit of the sequence if it exists

Special Types of Sequences 1, 3, 5, 7, 9, … 3, 6, 9, 12, 15, … What do you notice about each of these sequences? each of these sequences? They increase by a fixed amount for each subsequent member. for each subsequent member. 1, 3, 5, 7, 9,… = 1, 1 + 2, 1 + 4, 1 + 6, … a, a + d, a + 2d, a + 3d, … We call “d” the common difference.

Special Types of Sequences 1, 3, 5, 7, 9, … 3, 6, 9, 12, 15, … What is the common difference? 2 3

Special Types of Sequences 1, 3, 5, 7, 9, … Common difference = ? Arithmetic Sequence: a recursively defined sequence that that follows the pattern: that follows the pattern: a + (n – 1)d for n = 1 to ∞ n: #: 1, 3, 5, 7, 9, … a = 1 a = 1+(2-1)2 a = 1+(3-1)2 a = 1+(4-1)2 a = 1+(5-1)2 2

Special Types of Sequences 2, 5, 8, 11, 14, … Common difference = ? Arithmetic Sequence: a recursively defined sequence that that follows the pattern: that follows the pattern: a + (n – 1)d for n = 1 to ∞ n: #: 2, 5, 8, 11, 14, … a = 2 a = 2+(2-1)3 a = 2+(3-1)3 a = 2+(4-1)3 a = 2+(5-1)3 3

Arithmetic Sequence

Arithmetic Sequences Find (a)the common difference (b)the tenth term (c)a recursive rule for the nth term (d)an explicit rule for the nth term. -2, 1, 4, 7, … 3 For: n ≥ 2 Now simplify a + (n – 1)d for n = 1 to ∞

Your turn: Find 8. the common difference 9. the tenth term 10. recursive rule for the nth term 11. an explicit rule for the nth term. 4, 9, 14, 19, … a + (n – 1)d for n = 1 to ∞

Special Types of Sequences 2, 4, 8, 16, 32, … 2, 6, 18, 54, 162, … What do you notice about each of these sequences? each of these sequences? They increase by an increasing amount for each subsequent member. Geometic Sequence: a recursively defined sequence that that follows the pattern: that follows the pattern: We call “r” the common ratio. Recursively you can find the member of the sequence by multiplying the preceeding member by the common ratio. by multiplying the preceeding member by the common ratio.

Exp: (n-1) Geometric Sequences 2, 4, 8, 16, 32, … What is “r” (the common ratio)? Recursively 2 n: n: Explicitly Sequence: Sequence: Recursively Explicitly

Your turn: 2, 6, 18, 54, 162, … Recursive Explicit 12. What is the common ratio for the following sequence? 13. The first number in a sequence is 3. The common ratio is 4. What are the first 4 terms in the sequence? 14. The first number in a sequence is 3. The common ratio is 4. What is the 10 th number in the sequence? 3 10th term:

Geometric Sequence

Your turn: 15. Find the common ratio 16. Find an explicit rule for the nth term. 17. Find a recursive rule for the nth term 18. Find the tenth term 4, 20, 100,… r = 5

Another Famous Sequence 1, 1, 2, 3, 5, 8, 13, 21, … Can you define the pattern? cles/fibonac/index.asp F(1) = 1 F(2) = 1 F(n) = F(n – 1) + F(n – 2) The first two numbers are 1 and 1. Each number after that is the sum of the previous two numbers.

Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, … cles/fibonac/index.asp

Cool Nautilus shells growth using a pattern based upon this sequence. Other growing things do as well.

A Famous sequence. The numerators are the 2 nd and follow-on numbers in the Fibonacci sequence. numbers in the Fibonacci sequence. The denominators are the numbers in the Fibonacci sequence.

A Famous sequence. Does this converge? solve for “x” Converges to:

A Famous sequence. This is the “golden ratio” of the ancient Greeks They thought that the ratio was asthetically pleasing. They used it in their artwork, sculpture, and architecture. The “Golden Triangle, is an Isosceles triangle using this ratio.

Cool!

The Fibonacci Sequence