 # K12MATH011: Algebra II

Unit 9: Sequences and Series   In this final unit, you will be introduced to sequences and series. A sequence is a function defined over the positive integers. A series is a sequence in which the terms are summed. For general functions, the domain may be any subset of the real or complex numbers, but sequences are restricted to counting numbers. Sequences are often used to mimic functions in complex processes and are often used in mathematical modeling and analysis.

Unit 9 Time Advisory
Completing this unit should take approximately 10 hours and 30 minutes.

☐    Subunit 9.1: 1 hour and 15 minutes

☐    Subunit 9.2: 3 hours and 15 minutes

☐    Subunit 9.3: 2 hours

☐    Subunit 9.4: 45 minutes

☐    Subunit 9.5: 3 hours and 15 minutes

Unit9 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Find the nth term of any sequence, given the first four or five terms. - Find the nth term of an arithmetic sequence. - Find the nth term of a geometric sequence. - Find the sum of a finite geometric sequence using the sum formula.

9.1 Definition of Sequences and Series   Sequences are progressions of numbers that may be random but most often are formed by patterns. They are useful in that they may be used in programming to mimic certain functions, to which they are closely related.

• Explanation: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Introduction to Recursion and Sequences” Link: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Introduction to Recursion and Sequences” (HTML)

Instructions: This is a great description of sequences and recursive techniques. Sequences are often used to substitute for functions in some processes. Techniques for finding formulas lead to function creation. These techniques are used in physics and other natural sciences, business analysis and manufacturing, and for medical studies. To practice, try the randomly generated practice problems on the bottom of the page.

Completing this activity should take approximately 45 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Explicit and Recursive Definitions of Sequences” Link: Khan Academy’s “Explicit and Recursive Definitions of Sequences” (YouTube)

Instructions: This video discusses sequences in general and how to determine elements in a given sequence set.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Equations of Sequence Patterns” Link: Khan Academy’s  “Equations of Sequence Patterns” (YouTube)

Instructions: This video discusses a method for approximating a sequence via a formula or an equation. Patterns are often approximated by equations in the real world for purposes of prediction for the future or to make sense of a process for analysis.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

9.2 Arithmetic Sequences   These are sequences formed by starting with a value and adding the same fixed number over and over again. These sequences are related to and may be substituted for linear functions.

• Activity: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Arithmetic and Geometric Sequences” Link: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Arithmetic and Geometric Sequences” (HTML)

Instructions: This is a great description of arithmetic and geometric sequences. These sequences are used in physics and other natural sciences, business analysis and manufacturing, and for medical studies. Note that there are randomly generated practice problems on the bottom of the page. Use these for practice. After completing this task, write a one-page lesson comparing and contrasting arithmetic and geometric sequences, as if you were going to present the topic to your classmates. In particular, describe how they are different and how to develop a general formula for finding a particular term in each.

Completing this activity should take approximately 1 hour and 30 minutes.

Standards Addressed (Common Core):

Instructions: This video discusses the general pattern called an arithmetic sequence, formed by taking a starting value and adding the same value over and over to it. By learning how the pattern is formed, any general nth term of the sequence can be computed.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Arithmetic Sequences 1” Link: Khan Academy’s “Arithmetic Sequences 1” (HTML)

Instructions: This is a short self-assessment on arithmetic sequences. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Arithmetic Sequences 2” Link: Khan Academy’s “Arithmetic Sequences 2” (HTML)

Instructions: This is another short self-assessment on arithmetic sequences. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

9.3 Geometric Sequences   In geometric sequences, each term is found by taking a starting value and multiplying it repeatedly by a fixed, nonzero number. These sequences are among the most commonly used in mathematics, and they are related to and may substitute for exponential functions.

• Explanation: Khan Academy’s “Geometric Sequences Introduction” Link: Khan Academy’s “Geometric Sequences Introduction” (YouTube)

Instructions: This video discusses the general pattern called a geometric sequence, formed by taking a starting value and multiplying the same value over and over to it. By learning how the pattern is formed, any general nth term of the sequence can be computed.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

Instructions: This video continues the discussion of geometric sequences.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Geometric Sequences 1” Link: Khan Academy’s “Geometric Sequences 1” (HTML)

Instructions: This is a short self-assessment on geometric sequences. Hints are available, if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Geometric Sequences 2” Link: Khan Academy’s “Geometric Sequences 2” (HTML)

Instructions: This is another short self-assessment on geometric sequences. Note that there are hints available if needed.

Completing the practice problems and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

9.4 The Fibonacci Sequence   The Fibonacci sequence was named for Leonardo of Pisa, called Fibonacci, who first studied the pattern. The Fibonnaci sequence begins with 0 and 1. Then the sequence is generated by adding consecutive numbers to get the next term. Applications can be found in the fields of art, engineering, botany, computer programming, and business.

9.5 The Sum Formula   The sum of terms of a sequence is called a series. Such sums can almost always be patterned into a formula, provided the sequence is not merely a random set of numbers. This is one example, using the sum of arithmetic sequences, which can most definitely be used to create a formula.

• Explanation: Khan Academy’s “Sum of Arithmetic Sequence (Arithmetic Series)” Link: Khan Academy’s “Sum of Arithmetic Sequence (Arithmetic Series)” (YouTube)

Instructions: This video discusses a computation for the sum of the first n terms of an arithmetic sequence.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Explanation: Khan Academy’s “Formula for Arithmetic Series” Link: Khan Academy’s “Formula for Arithmetic Series” (YouTube)

Instructions: This video demonstrates how the formula for an arithmetic series is created and shows how to use it.

Watching the video and writing notes should take approximately 15 minutes.

Standards Addressed (Common Core):

• Did I Get This? Activity: Khan Academy’s “Arithmetic Series” Link: Khan Academy’s “Arithmetic Series” (HTML)

Instructions: This is a self-quiz on arithmetic series. Note that there are hints available if needed, and there is a calculator on the page as well.

Taking the quiz and writing notes should take approximately 45 minutes.

Standards Addressed (Common Core):

• Checkpoint: Mathematics Vision Project’s “3.2 Sequences” Link: Mathematics Vision Project’s “3.2 Sequences” (PDF)

Instructions: Scroll down to page 8. This is an assessment on arithmetic sequences. Complete the “Ready, Set, Go” problems 1-14.

Completing this assessment should take approximately 30 minutes.

Standards Addressed (Common Core):

• Checkpoint: Mathematics Vision Project’s “3.7 Sequences” Link: Mathematics Vision Project’s “3.7 Sequences” (PDF)

Instructions: Scroll down to page 28. This is an assessment on geometric sequences. Complete the “Ready, Set, Go” problems 1-16.

Completing this assessment should take approximately 30 minutes.

Standards Addressed (Common Core):

• Checkpoint: MathPortal: “Tests in Sequences and Series” Link: MathPortal: “Tests in Sequences and Series” (HTML)

Instructions: This page contains three 10-question tests on sequences and series. Work all problems. Answers are available for viewing when each test is completed.

Completing this assessment should take approximately 30 minutes.

Standards Addressed (Common Core):

• Checkpoint: Khan Academy: “Recursive and Explicit Functions” Link: Khan Academy: “Recursive and Explicit Functions” (HTML)

Instructions: This is an assessment on recursive and explicit functions. Work all problems. Answers are available for viewing by a button on each problem.

Completing this assessment should take approximately 15 minutes.

Standards Addressed (Common Core):