Unlocking Arithmetic Sequences: Mastering Recursive Formulas

Algebra 2 Grades High School 4:38 Video
Print Lesson Plan Watch Video

Lesson Description

Learn how to define arithmetic sequences using recursive formulas, building on your knowledge of explicit formulas. This lesson provides a step-by-step guide with examples, ideal for Algebra 2 students.

Video Resource

Arithmetic Recursive

Kevinmathscience

Duration: 4:38
Watch on YouTube

Key Concepts

  • Arithmetic Sequence
  • Recursive Formula
  • Common Difference

Learning Objectives

  • Define arithmetic sequences and identify the common difference.
  • Express arithmetic sequences using recursive formulas.
  • Use recursive formulas to find specific terms in a sequence.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of arithmetic sequences and their patterns (adding or subtracting a constant value). Briefly recap explicit formulas. State the learning objective: understanding and using recursive formulas.
  • Understanding Recursive Formulas (10 mins)
    Explain what a recursive formula is: a formula that defines a term based on the previous term(s). Contrast this with an explicit formula. Use the example from the video (3, 5, 7, 9) to demonstrate how to write the recursive formula: a(n) = a(n-1) + 2. Emphasize that the recursive formula needs an initial value (a(1)).
  • Example Problems (15 mins)
    Work through the examples from the video, showing step-by-step how to write the recursive formula for sequences with both addition and subtraction (20, 17, 14, 11 and 1.5 increments). Reinforce the notation a(n) and a(n-1). Introduce a new example: 1, 4, 7, 10... What is the recursive formula? Have students attempt to determine the formula before revealing the answer. Discuss the common difference.
  • Applying the Formula (10 mins)
    Present the example where the recursive formula is given (a(n) = a(n-1) + 3, a(1) = 4). Guide students through finding the first four terms of the sequence. Emphasize the importance of the initial term. Ask guiding questions: "What does a(1) represent?", "How do we find a(2) given a(1)?".
  • Practice and Review (5 mins)
    Assign practice problems (see interactive exercises below). Review key concepts and address any remaining questions.

Interactive Exercises

  • Write the Recursive Formula
    Provide students with several arithmetic sequences (e.g., 2, 6, 10, 14...; 15, 12, 9, 6...). Ask them to write the recursive formula for each sequence, including the initial term.
  • Find the Terms
    Give students a recursive formula and an initial term (e.g., a(n) = a(n-1) - 5, a(1) = 20). Ask them to find the first five terms of the sequence.

Discussion Questions

  • What are the advantages and disadvantages of using a recursive formula versus an explicit formula?
  • Can all sequences be defined by a recursive formula? Why or why not?
  • How does the initial term affect the entire sequence when using a recursive formula?

Skills Developed

  • Pattern Recognition
  • Abstract Reasoning
  • Formula Application

Multiple Choice Questions

Question 1:

Which of the following best describes a recursive formula?

Correct Answer: A formula that defines a term based on the previous term(s).

Question 2:

What is the 'common difference' in an arithmetic sequence?

Correct Answer: The constant value added or subtracted to get to the next term.

Question 3:

Given the recursive formula a(n) = a(n-1) + 4, a(1) = 1, what is the value of a(3)?

Correct Answer: 9

Question 4:

In a recursive formula, what does a(n-1) represent?

Correct Answer: The term before the nth term.

Question 5:

Which of the following sequences could be defined using a recursive arithmetic formula?

Correct Answer: 2, 5, 8, 11...

Question 6:

Which component is essential for defining an arithmetic sequence recursively?

Correct Answer: The initial term

Question 7:

What is the recursive formula for the sequence 8, 5, 2, -1, ...?

Correct Answer: a(n) = a(n-1) - 3, a(1) = 8

Question 8:

If a(n) = a(n-1) + d, what does 'd' represent in this recursive formula for an arithmetic sequence?

Correct Answer: The common difference

Question 9:

Which statement regarding arithmetic and geometric sequences is always true?

Correct Answer: An arithmetic sequence has a common difference; a geometric sequence has a common ratio.

Question 10:

What must always be provided to initiate a recursive sequence?

Correct Answer: Both the recursive formula and the first term.

Fill in the Blank Questions

Question 1:

A formula that defines a term based on the previous term(s) is called a ________ formula.

Correct Answer: recursive

Question 2:

In an arithmetic sequence, the constant value added or subtracted to get to the next term is called the ________ ________.

Correct Answer: common difference

Question 3:

To use a recursive formula, you must know the ________ ________ of the sequence.

Correct Answer: initial term

Question 4:

If a(n) = a(n-1) - 2 and a(1) = 8, then a(2) = ________.

Correct Answer: 6

Question 5:

The recursive formula for the sequence 5, 10, 15, 20... is a(n) = a(n-1) + ________, a(1) = 5.

Correct Answer: 5

Question 6:

An arithmetic sequence is a sequence where the difference between consecutive terms is always ________.

Correct Answer: constant

Question 7:

If a sequence follows the recursive rule a(n) = a(n-1) + 7 and the first term a(1) is -3, then the second term a(2) is ________.

Correct Answer: 4

Question 8:

In the recursive formula a(n) = a(n-1) + d, 'a(n)' represents the ________ term of the sequence.

Correct Answer: nth

Question 9:

The initial term of an arithmetic sequence, along with the ________ ________, completely defines the sequence.

Correct Answer: recursive formula

Question 10:

For the recursive definition of an arithmetic sequence to be complete, both the formula and the ________ term must be specified.

Correct Answer: first

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.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans