Unlocking Recursive Formulas: A Deep Dive into Sequences

Algebra 2 Grades High School 7:18 Video
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Lesson Description

Master the art of writing recursive formulas for arithmetic, geometric, and Fibonacci sequences. Learn how to define the first term and derive formulas for subsequent terms.

Video Resource

Recursive Formulas How to Write

Mario's Math Tutoring

Duration: 7:18
Watch on YouTube

Key Concepts

  • Recursive formulas
  • Arithmetic sequences
  • Geometric sequences
  • Fibonacci sequence
  • Explicit vs. Recursive Formulas

Learning Objectives

  • Students will be able to define and write recursive formulas for arithmetic sequences.
  • Students will be able to define and write recursive formulas for geometric sequences.
  • Students will be able to convert from an explicit formula to a recursive formula.
  • Students will be able to identify and write the recursive formula for the Fibonacci sequence.

Educator Instructions

  • Introduction (5 mins)
    Begin by introducing the concept of recursive formulas and contrasting them with explicit formulas. Briefly explain their utility in computer programming and repetitive calculations. Preview the types of sequences to be discussed (arithmetic, geometric, Fibonacci).
  • Arithmetic Sequences (10 mins)
    Analyze Example 1 (3, 7, 11, 15, 19...). Guide students to identify the common difference. Demonstrate how to write the recursive formula, emphasizing the definition of the first term (a_1) and the recursive step (a_n = a_(n-1) + d). Provide additional examples for students to practice.
  • Geometric Sequences (10 mins)
    Analyze Example 2 (5, 10, 20, 40, 80...). Guide students to identify the common ratio. Demonstrate how to write the recursive formula, emphasizing the definition of the first term (a_1) and the recursive step (a_n = a_(n-1) * r). Provide additional examples for students to practice.
  • Explicit to Recursive (10 mins)
    Explain how to derive a recursive formula from a given explicit formula (Examples 3 & 4). Emphasize the strategy of calculating the first few terms of the sequence to identify the pattern. Show how the common difference or ratio in the explicit formula translates into the recursive formula.
  • Fibonacci Sequence (10 mins)
    Introduce the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...). Explain the unique recursive definition where each term is the sum of the two preceding terms. Highlight the need to define the first two terms (a_1 and a_2) in this case.
  • Wrap-up and Practice (5 mins)
    Summarize the key concepts covered in the video. Assign practice problems for homework or in-class activity to reinforce understanding. Encourage students to explore more complex sequences and their recursive representations.

Interactive Exercises

  • Sequence Identification
    Present students with a series of sequences (arithmetic, geometric, neither). Have them identify the type of sequence and, if applicable, determine the common difference or ratio.
  • Recursive Formula Writing
    Provide students with explicit formulas for sequences. Challenge them to derive the corresponding recursive formulas.

Discussion Questions

  • What are the key differences between explicit and recursive formulas?
  • In what situations would a recursive formula be more useful than an explicit formula?
  • Can you think of real-world examples of arithmetic, geometric, or Fibonacci sequences?

Skills Developed

  • Pattern recognition
  • Abstract reasoning
  • Mathematical notation
  • Problem-solving

Multiple Choice Questions

Question 1:

Which of the following sequences is arithmetic?

Correct Answer: 1, 5, 9, 13...

Question 2:

What is the common ratio in the geometric sequence 3, 6, 12, 24...?

Correct Answer: 2

Question 3:

In a recursive formula, a_n represents:

Correct Answer: The nth term

Question 4:

What are the first two terms defined as in the recursive formula for the Fibonacci Sequence?

Correct Answer: a_1=1, a_2=1

Question 5:

Which formula best represents an arithmetic sequence?

Correct Answer: a_n=a_(n-1)+d

Question 6:

In the Fibonacci sequence, what is the next term after 5 and 8?

Correct Answer: 13

Question 7:

Which formula best represents a geometric sequence?

Correct Answer: a_n=a_(n-1)*r

Question 8:

What is the value of the second term given the explicit formula a_n = 4n -3, and the recursive formula a_n= a_(n-1) +4?

Correct Answer: 5

Question 9:

A recursive formula requires knowing the value of the __________ term to find the next term.

Correct Answer: Previous

Question 10:

When converting from explicit to recursive, what is the most efficient first step?

Correct Answer: Calculate the first few terms of the sequence

Fill in the Blank Questions

Question 1:

A __________ formula allows you to calculate any term directly, whereas a recursive formula builds upon previous terms.

Correct Answer: explicit

Question 2:

In an arithmetic sequence, you add the __________ __________ to get the next term.

Correct Answer: common difference

Question 3:

In a geometric sequence, you multiply by the __________ __________ to get the next term.

Correct Answer: common ratio

Question 4:

The Fibonacci sequence is defined by adding the two __________ terms together.

Correct Answer: previous

Question 5:

The recursive formula for an arithmetic sequence is written as a_n = a_(n-1) + __________.

Correct Answer: d

Question 6:

The recursive formula for a geometric sequence is written as a_n = a_(n-1) * __________.

Correct Answer: r

Question 7:

To define a recursive formula, you must always state the value of the __________ term(s).

Correct Answer: first

Question 8:

Recursive formulas are especially useful in __________ programs for performing repetitive calculations.

Correct Answer: computer

Question 9:

When writing the recursive formula for the Fibonacci sequence, you go __________ terms back.

Correct Answer: two

Question 10:

The nth term is also known as a________.

Correct Answer: sub n

Educational Standards

A2.A.3:Represent and solve mathematical and real-world problems involving arithmetic and geometric sequences and series. A2.A.3.1:Recognize that arithmetic sequences are linear using equations, tables, graphs, and verbal descriptions. Using the pattern, find the next term. A2.A.3.2:Recognize that geometric sequences are exponential using equations, tables, graphs, and verbal descriptions. Given the formula 𝑓(𝑥) = 𝑎(𝑟)ˣ, find the next term and define the meaning of 𝑎 and 𝑟 within the context of the problem.

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