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Understanding Arithmetic and Geometric Sequences: Patterns and Formulas

This lesson focuses on recognizing and constructing arithmetic and geometric sequences. Begin by assessing your understanding through a guided anticipation exercise. After reading textbook pages 28-31, refine your responses based on newly acquired knowledge. We will discuss recursive formulas, common differences, ratios, and shifted geometric sequences. You'll practice identifying patterns and generating sequences from equations. Finally, apply your understanding through homework worksheets and prepare for an upcoming quiz.

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Understanding Arithmetic and Geometric Sequences: Patterns and Formulas

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  1. Warm Up • Find the Pattern: • 130, 65, 32.5, 16.25 • 46, 55.5, 65, 74.5 • 6, 12, 24, 48

  2. Answers • *1/2 • +9.5 • *2

  3. Directions Step 1: Start the anticipation guide by filling out the “Me” column with true or false based on your current ideas. Step 2: Read pages 28-31 in your textbook Step 3: Go back and complete the “Text” column with your new answers after reading about the topic.

  4. Investigation Page 31

  5. LECTURE • Recursive Formulas: 1.1

  6. Notation • When describing sequences we use “U sub n”. is the term you are talking about, and is the previous term. • This is weird at first, but you will get used to it.

  7. Definitions • Arithmetic Sequence • The previous term plus a constant. • The constant is called the “common difference” • How do we know if it is arithmetic? • should be the same number for each

  8. Definitions • Geometric Sequence: • The previous term multiplied by a constant • Constant is called a “common ratio” • How do we know it is geometric? should be the same number for each pair.

  9. Definition • Shifted Geometric Sequence • Combines arithmetic and geometric. • We will look at these more in a couple days.

  10. To create a sequence given an equation, you need two things. • The first number of the sequence. 2. The equation.

  11. Example 1 Arithmetic or geometric? Sequence is: {2,5,8,11,14…}

  12. Example 2 Arithmetic or geometric? Sequence is: {1,3,9,27…}

  13. Example 3 • To go from a sequence to an equation, you have to find the pattern (look at your definitions)

  14. Name that sequence!Arithmetic or Geometric? Arithmetic Geometric Arithmetic Geometric • 6, 11, 16, 21… • 100, 10, 1, 0.1… • -4, -3.99, -3.98… • .0625, .125, .25

  15. You try a couple… Generate the sequence for each equation below.

  16. Example 4 • 5000, 500, 50, 5 • Clearly this is not arithmetic, so now we check for the common ratio. • 500/5000=0.1 • 50/500=0.1 • 5/50=0.1 All the same, so the common difference is .1. Equation is

  17. You try a few • 6, 11, 16, 21… • 100, 10, 1, 0.1… • .0625, .125, .25 +5 *0.1 *2

  18. Homework • Worksheet 1.1 • Skip 6. *This is a big assignment, use your time well! ** Quiz on Monday!

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