Finding Patterns of Sequences and Graphing (Day 1)
A sequence is a set of ______written in a given ______.
Write the next three termsof each sequence, then describe the pattern:
1.4, 8, 12, 16, …
2.3, 9, 27, 81, …
3. 1, 4, 9, 16, …
Now, come up with your own patterns for number 4 and 5:
4.5.
Find the 10th term of each of the five sequences above.
Now observe the tables below and fill in the missing cells:
Term Number / Term1 / 4
2 / 8
3 / 12
4 / 16
Sequence Term / Term
a1 / 4
a2 / 8
a3 / 12
a4 / 16
New Notation:
f(1)
f(2)
f(3)
f(4)
f(n) =
What is the name of this new notation? How do you say it and what does it mean?
Consider the sequence from the previous page. Graph the coordinates from the table.
n / f(n)1 / 4
2 / 8
3 / 12
4 / 16
What is the name of this type of function? Write the equation of this function.
Consider this sequence and complete the table. Graph the coordinates from the table.
n / f(n)1 / 2
2 / 4
3 / 8
4 / 16
What is the name of this type of function? Write the equation of this function.
Let’s now use the functions from the previous page.
f(n) = 4nf(n) = 2n
Find the following and explain in words what it means.
1. f(6) = 3.f(7) =
2.f(50) = 4.f(10) =
Summary:
1.Why is it important for a sequence to have a formula?
2.What does f(n) represent? How is it read aloud?
3.How do you represent the terms of a sequence on a graph?
Artihmetic vs. Geometric Sequences (Day 2)
Recall: 1. Fill in the rest of the table with the next 2 terms of the sequence.
n / f(n)1 / 5
2 / 10
3 / 15
4 / 20
2.What is the pattern of the sequence?
3.Write the equation of the function.
4. What type of function is this?
5.Find f(25).
6.What is the form of the coordinates that come from the sequence?
Types of Sequences:
- Arithmetic Sequence: a set of ______in a specific order each having a ______.
- Example:1, 6, 11, 16, 21,…(common difference = ______)
- Example:15, 11, 7, 3, -1,…(d = ______)
- Geometric Sequence: a set of ______in a specific order having a ______.
- Example:1, 2, 4, 8, 16,…(common ratio = ______)
- Example:16, -8, 4, -2, 1,…(r = ______)
In 1 – 4, a sequence is described.
a)Determine if they are geometric, arithmetic, or neither.
b)Give a reason for your choice.
1.5, 10, 15, 20, 25, …3.10, 30, 90, 270, 810, …
2.64, 32, 16, 8, 4, …4.1, 4, 9, 16, 25, …
Now consider the sequences that we observed on Day 1. Fill in the rest of the tables and think about the patterns. Answer the following questions.
1.Does the sequence have a common difference or ratio? What is it?
2.Is the sequence arithmetic or geometric?
3.Write the formula for the function.
n / f(n)1 / 4
2 / 8
3 / 12
4 / 16
n / f(n)
1 / 2
2 / 4
3 / 8
4 / 16
All arithmetic sequences are ______functions.
All geometric sequences are ______functions.
Practice:
Given the following sequences, answer the following:
a.What are the next three terms?
b.What type of sequence is it and why?
c.What is the formula of the function?
d.What type of function does the sequence represent?
e.What is the 20th term of the sequence?
1.4, -1, -6, -11,…2.3, -1, 1/3, -1/9, …
Explicit vs. Recursive Formulas (Day 3)
Recall:Given the sequence -2, 2, 6, 10, …, find:
a.What are the next three terms?
b.What type of sequence is it and why?
c.What is the formula of the function?
d.What type of function does the sequence represent?
e.What is the 100th term of the sequence?
Explicit Formula:A formula that can solve for ______term of a sequence.
Is the formula we found for the sequence above explicit? Why?
For the sequence 5, 8, 11, 14, …, Anthony came up with the formula A(n) = 5 + (n – 1)3.
Is his formula correct and how do you know? Is it an explicit formula?
What does A(n) mean?
For the same sequence, Johnny wrote the formula as J(n) = J(1) = 5
J(n + 1) = J(n) + 3
What does J(1) mean?
What does J(n + 1) mean?
Can you find J(5) using formula J? Why?
Can you find J(100) using formula J? Why?
Recursive Formula: A formula that allows any term a(n) to be computed from the ______term. You must be given the ______term a(1) to get you started.
So, is Johnny’s formula recursive or explicit?
Practice:Answer the following parts for each question below:
a.Translate these formulas into words.
b.Are the formulas representing a sequence that is arithmetic or geometric?
c.Find the next three terms of the sequence.
1.B(1) = 12
B(n + 1) = B(n) – 3
2.P(1) = 7
P(n + 1) = -2P(n)
There is another notation that you need to know about. What do these mean below?
A(n) is the same as an. A(1) is the same as a1. A(50) is the same as a50.
So, a subscript is the same as ______.
For #3 – 4, write the two formulas above with the new notation (using subscripts).
3.4.
5.For the sequence 12, 7 , 2, -3, …
a)Write an explicit formula.
b) Write a recursive formula.
Another way of writing this recursive formula is:
In 6-9, a sequence is described.
a)Identify the definition as recursive or explicit.
b)Find the first three terms of the sequence.
6.r1 = -47.t(n) = -6n3 + 2n2
rn = -3rn-1
8.j1 = 39.kn = 12 – 3(n – 1)
jn+1 = (jn)2
10.Write both a recursive and an explicit definition for the following sequence:
4, 8, 12, 16, 20
11.Given the sequence:2, 4, 8, 16,…
a)List the next three terms of the sequence.
b)Write an explicit formula for an.
c)Write a recursive definition for the sequence.
Sequences Review (Day 4)
Arithmetic Sequence:
Geometric Sequence:
Explicit Formula:
Recursive Formula:
For the following sequences, answer…
a) What are the next three terms.
b)Are the following arithmetic or geometric and why?
c)Write an explicit formula.
d)Write a recursive formula.
e)Use a formula to find the a(9).
1.14, 11, 8, 5, …2. 1, 10, 100, 1000, …
3.14, 21, 28, 35, …4.2, 10, 50, 250,…
As you saw from the last exercise, writing an explicit formula can sometimes be extremely difficult. Well, here is something that will help…
5.For the sequence 100, 97, 94, 91, …, find:
a)the common difference.
b)an explicit formula for the sequence.
c)the 20th term of the sequence.
6.The local football team won the championship several years ago, and since then, ticket prices have been increasing $4 per year. The year they won the championship, tickets were $20. Write anexplicit formula for a sequence that will model ticket prices. What will the ticket price be 10 years after that championship year?
7.Cooper is saving to buy a guitar. In the first week, he put aside $10 that he received for his birthday. In each of the following weeks, he decided he will add put aside $8 more than the week before. How much money will he be putting aside in the 7th week?
8.Given the sequence:4, 12, 36, 108, 324, …
a)What is the common ratio?
b)Write an explicit formula for this sequence.
c)What is the 10th term of this sequence?
9.What is the 8th term of the geometric sequence 125, 25, 5,…?
These formulas will be given to you on the reference table on the final exam. So…you will be given them for any quizzes or tests!!
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