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Arithmetic and Geometric Sequences: Definitions, Formulas, and Analysis

Learn about arithmetic and geometric sequences, their recursive definitions, explicit formulas, and how to analyze and find terms in these sequences.

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Arithmetic and Geometric Sequences: Definitions, Formulas, and Analysis

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  1. For each figure, how is the number on the center tile related to the numbers on the other tiles? What will the center number in Figure 6? What will the center number be in figure 10?

  2. Warm Up Simplify and state restrictions. , =

  3. Sequences and Series Unit Objectives: Describe a list of numbers using sequence/series terminology Write recursive definitions, explicit formulas and summation notation for sequences/series Find values for arithmetic/geometric sequences/series. Model problems using sequences/series 9-2 Today’s Objective: I can define, identify and apply arithmetic sequences.

  4. Sequences Term of a Sequence: Sequence: Each number: n represents term number Ordered list of numbers Recursive Definition:Uses the previous term () Two Parts: Initial ValueRecursive Rule Explicit Formula: Describes sequence using term number (n)

  5. Arithmetic Sequence 4, 7, 10, 13, 16, … a, a + d, a + 2d, a + 3d, … +3 +3 +3 +3 a = starting value Recursive Definition: d = common difference Recursive Definition: for Explicit Formula: 1, 4, 9, 16, 25, … Explicit Formula: for 5 9 7 3 Not an Arithmetic Sequence

  6. Analyzing Arithmetic Sequences Find the 2nd and 3rd term of:100, ▒ , ▒, 82, … Explicit Formula: Find the 46th term:3, 5, 7, … 94, 88, Find the 24th term: 4, 7, 10, … Finding missing term: …, 15, ▒ , 59, … 37, Arithmetic Mean: …, a, b, c, …

  7. 9-3Geometric Sequences Today’s Objective: I can define, identify and apply geometric sequences.

  8. Geometric Sequence 3, 6, 12, 24, 48, … a, a∙r, a∙r2, a∙r3, … a = starting value r= common ratio: Recursive Definition: Explicit Formula: Recursive Definition: , for n > 1 2, 8, 32, 128, … Explicit Formula: , for n≥ 1

  9. Analyzing Geometric Sequences Geometric Mean: …, a, b, c, . . . Find the 10thterm:4, 12, 36, … Find the 2nd and 3rd term of:2, ▒ , ▒ , − 54, … 18, – 6, Explicit Formula: Explicit Formula: Finding the possible missing term: …, 48, ▒ , 3, … ±12,

  10. Sierpinski Triangle p. 575: 7-23 odd, 41-49 odd p. 584: 7-17 odd, 33-43 odd Stage 4 Stage 1 Stage 2 Stage 3 How many red triangles are there at stage 20? 20 1,162,261,467 1 3 9 27 Recursive Definition: Explicit Formula:

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