Sequences

Standard: A.F.IF.3 "Recognize that sequences are functions, sometimes defined recursively, whose domain is the subset of the integers."

Essential Question:

How do I recognize that sequences are functions?

Write the next three terms in each pattern and explain how you generated each term.(Hint: When letters are used, think about what each letter could represent.)

1)  J, F, M, A, M, J, J, A, S, ...

2) S, M, T, W, ...

3)  5, 10, 15, 20, ...

4)  100, 81, 64, 49, ...

Vocabulary:

1. Sequence - a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects.

2. finite sequence - a sequence that terminates

3. infinite sequence - a sequence which continues forever

4. term of a sequence - an individual number, figure, or letter in a sequence

I'll pass out these new books. Turn to page 214. DO NOT WRITE IN THEM!!

Class work assignment: pg. 51-52 of your workbook

1. Analyze Pascal's Triangle.a. Describe the pattern of the number of terms in each row.

b. Describe the pattern of each row.

c. Describe the pattern that results from determing the sum of each row.

d. Determine the next two rows in Pascal's Triangle.

2. Analyze the diagonals labeled on Pascal's Triangle.

Student text time again! Start on pg. 215 with Al's Omletes (linear). Then do Mario's Mosaic (exponential), Troop of Triangles, and Gamer Guru. You can work with a partner, but NOT A GROUP! You have 25 minutes.

Homework assignment: pg. 333-337, #'s 1-16 all

Essential Question:

How do I find the common difference in a sequence and recognize whether the sequence is arithmetic or geometric?

Is the sequence infinite or finite?

Student text pg. 220 - we will walk through this together.

(Fibonacci Sequence...)

Write the next three terms in each sequence and explain how you generated each term:

a. -13, -8, -3, 2, 7,...

b. -2, 4, -8, 16,...

c. 60, 53, 46, 39, 32,...

d. 1, 5, 17, 53, 162, 488 (you have to do 2 things to each term)

Arithmetic SequenceDefinition of Arithmetic Sequence Arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms. In other words, arithmetic sequence is a sequence of numbers in which each term except the first term is the result of adding the same number, called the common difference, to the preceding term.Example of Arithmetic Sequence The sequence 5, 11, 17, 23, 29, 35 . . . is an arithmetic sequence, because the same number 6 (i.e. the common difference) is added to each term of the sequence to get the succeeding term.

Geometric SequenceDefinition of Geometric Sequence Geometric sequence is a sequence in which each term after the first term a is obtained by multiplying the previous term by a constant r, called the common ratio. It is obvious that a ≠ 0 and r ≠ 0 or 1.1, 2, 4, 8, 16, 32, . . . is a geometric sequence.Each term of this geometric sequence is multiplied by the common ratio 2.

Bottom Line: if a sequence is arithmetic, you add or subtract; if a sequence is geometric, you multiply or divide.

Finding the "common difference" (or constant) is the key to recognizing the pattern.

Turn to pg. 225 and do A through D quickly.

DIRECTIONS: finish the sequence, state the common difference, and tell me whether it is arithmetic or geometric.

**Hint** D is challenging, see if you can figure it out.

Class work / Homework:workbook pg. 339-341, #'s 1-40 ALL

GET STARTED NOW!!!

1. Arithmetic: adding 10 each time2. Geometric: multiplying by 2 each time

Problem Set:2. -7 3. 2.5 4. 1/3 5. -3.5 6. 70

7. -60 8. -1.2 9. 1/2 10. 15

12. 4 13. -2 14. 1/2 15. -3

16. -1/2 17. 8 18. 1/3 19. -520. 1/5

22. 30, 15, 0 23. 16, 26, 36

24. 7/5, 8/5, 9/5 25. -16, -25, -34

26. 30, 34.5, 39 27. -145, -156, -167

28. 9, 10.3, 11.6 29. 1000, 1375, 1750

30. 9.3, 5.5, 1.7 32. 32, 16, 8

33. 80, -160, 320 34. .3, .03, .003

35. 2, -2, 2 36. 259.2, 1555.2, 9331.2

37. -500, 250, -125 38. 25.6, 102.4, 409.6

39. .25, .05, .01 40. 567, -1701, 5103

Essential Question: How do I use formulas to determine unknown terms of a sequence?

Standards: A.F.IF.3 / A.F.BF.1a / A.F.BF.2 / A.F.LE.2

Vocabulary:

Explicit Formula: calculates each term of a sequence using the term's position in the sequence

Recursive Formula: expresses each term of a sequence based on the preceding term of the sequence

Index: the position of a term in a sequence

Example 1:

Scott owns a sporting goods store. He has agreed to donate $125 to the Glynn Academy baseball team for their equipment fund. In addition, he will donate $18 for every home run the Red Terrors hit during the season. The sequence shown represents the possible dollar amounts that Scott could donate for the season:

125, 143, 161, 179,...a. Identify the sequence type - EXPLAIN how you know:

b. Determine the common ratio or difference for the given sequence:

c. Complete the table of values. Use the number of home runs the Red Terrors could hit to identify the term number, and the total dollar amount Scott could donate to the baseball team:

# of HR

Term #

$

0 1 $125

1

2

3

4

5

67

8

9

d. Explain how you would calculate the 10th term based on the 9th term:

e. Determine the 20th term. Explain your calculation.

f. Is there a way to calculate the 20th term without knowing or calculating the 19th term first?

g. The Red Terrors hit 93 homeruns. Calculate Scott's total donation. Explain your answer.

General Rule ExampleA lowercase letter is used to name a sequence.

a

The first term, or initial term, is referred to as a1.

a1 = 1

The remaining terms are named according to the term number.

a2 = 143, a3 = 161,...

A general term of the sequence is referred to as an, also known as the nth term, where n represents the index.

an

The term previous to an is referred to as an-1.

an-1

The common difference is represented as d.

d = 18

Analyze the table. The examples shown are from the sequence showing Scott's contribution to the Red Terror baseball team in terms of HR's hit:

an

From these rules we can develop a formula so that you do not need to determine the value of the previous term to determine subsequent terms.

an = a1 + d(n - 1)

* You must know the value of the first term AND the common difference*

term you want

first term

common differenceprevious term

So, if a1 = 125 and d = 18, find the following donation amount for:(SHOW YOUR WORK!) **The term # doesn't match the HR #!!!**a. 35 HR's

b. 48 HR's

c. 86th termd. 214th term

Example 2:Scott decides to change his donation amounts. He decides to contribute $500 and will donate $75 for every home run hit. Determine Scott's contribution if the Red Terrors hit:(Write the explicit formula FIRST!)

a. 11 HR's b. 26 HR's

c. 39th term d. 50th term

Homework:

pg. 343-344, #'s 1-3 in vocab section AND 1-10 in the problem set section.

1. recursive 2. index3. explicit

Problem Set:1. 58 2. -155 3. 29.7

4. -292 5. 104.50 6. -485

7. 98.7 8. 895 9. 0

10. 7200

Essential Question: How do I use formulas to determine unknown terms of a sequence?

Standards: A.F.IF.3 / A.F.BF.1a / A.F.BF.2 / A.F.LE.2

Example 1:

During growth, a virus cell, called a mother cell, divides itself into two daughter cells. Each of those cells divide into two more, and so on:

1, 2, 4, 8, 16,...

a. Identify the sequence type - EXPLAIN how you know:

b. Determine the common ratio or difference for the given sequence:

c. Complete the table of values. Use the number of cell divisions to identify the term number, and the total number of cells after each division:

# of Cell Divisions

Term #

Total # of Cells

0 1

1

2

3

4

5

67

8

9

d. Explain how you would calculate the 10th term based on the 9th term:

e. Determine the 20th term. Explain your calculation.

f. Is there a way to calculate the 20th term without knowing or calculating the 19th term first?

g. Calculate the total number of cells for the 53rd division. Then, calculate the total number of cells for the 53rd TERM.

General Rule ExampleA lowercase letter is used to name a sequence.

g

The first term, or initial term, is referred to as g1.

g1 = 1

The remaining terms are named according to the term number.

g2 = 2, g3 = 4,...

A general term of the sequence is referred to as gn, also known as the nth term, where n represents the index.

gn

The term previous to gn is referred to as gn-1.

gn-1

The common difference is represented as d.

r = 2

From these rules we can develop a formula so that you do not need to determine the value of the previous term to determine subsequent terms.

gn = g1 rn-1

* You must know the value of the first term AND the common ratio*

term you want

first term common ration

previous term number

So, if g1 = 1 and r = 2, answer the following questions:

(Show your work!)

Use the explicit formula for geometric sequences to determine the total number of cells:

1. after 11 divisions 2. after 14 divisions

3. Find the total # of cells for the 19th term

4. Find the total # of cells for the 22nd term

Example 2:

Suppose that a scientist has 5 virus cells in a petri dish. She wonders how the growth pattern would change if each mother cell divided into 3 daughter cells. For this situation, determine the total number of cells in the petri dish after:

1. 4 divisions 2. 16 divisions

3. The 13th term 4. The 7th term

Class work / Homework: pg. 344-345, #'s 11-20

11. 1536 12. 16,384 13. 885,73514. 294,912

15. -65,536 16. 387,420,489 17. 32,768

18. 195,312.5 19. 46.57 20. -9,765,625

Essential Question: How do I use formulas to determine unknown terms of a sequence?

Standards: A.F.IF.3 / A.F.BF.1a / A.F.BF.2 / A.F.LE.2

The explicit formula is just ONE WAY to determine terms of a sequence. There is another way, called the recursive formula.

A recursive formula expresses each new term of a sequence based on the preceding term in the sequence.

The recursive formula for determining the nth term of an arithmetic sequence is:

an = an-1 + d

**Please note: the difference between the explicit formula and recursive formula for an arithmetic sequence is...

Only use the recursive formula when looking for the NEXT term!

term you want

previous term common difference

The recursive formula for determining the nth term of a geometric sequence is:

gn = gn-1 r

**Please note: the difference between the explicit formula and recursive formula for an geometric sequence is...

Only use the recursive formula when looking for the NEXT term!

term you want

previous termcommon ratio

Determine whether each sequence is arithmetic or geometric. Then use the RECURSIVE FORMULA to determine the unknown term in each sequence:

1. 5/3, 5, 15, 45, _____...

2. -45, -61, -77, -93, _____,...

3. -3, 1, ____, 9, 13...

4. -111, 222, ____, 888, -1776...

5. -30, -15, ____, -3.75, -1.875, _____,...

6. 3278, 2678, 2078, _______, _______, _______,...

Consider this sequence again:

3278, 2678, 2078, 1478, 878, 278,...

Use the recursive formula (show your work) to find the 9th term of the sequence (note: you have to find the 7th and 8th term first!)

Now in one step, use the EXPLICIT FORMULA to determine the 9th term of the same sequence (show your work)!

3278, 2678, 2078, 1478, 878, 278,...

Which formula do you prefer? Justify your answer...

Which formula would you use if you were given 5 terms of a sequence and asked for the 6th?

Which formula would you use if you were given 5 terms of a sequence and asked for the 61st term?

HW: Do pg. 346-347, #'s 21-28.