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Understanding Arithmetic and Geometric Sequences: Formulas and Examples

This comprehensive guide covers arithmetic and geometric sequences, explaining how to find constant differences, explicit and implicit formulas, and the common ratio in geometric sequences. You'll learn how to evaluate specific terms in these sequences and apply formulas for given variables. Examples include identifying arithmetic sequences (e.g., 1, 5, 9, 13) and geometric sequences (e.g., 1, 3, 9, 27) and their properties. Exercises are included to test your understanding of the topic.

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Understanding Arithmetic and Geometric Sequences: Formulas and Examples

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  1. Do Now • Determine the constant difference and the explicit formula of arithmetic sequence of 1, 5, 9, 13… • Determine the implicit formula of • Find the values of a and d if we know that a10 = 38 and a15 = 18. 1.6: Geometric Sequences

  2. Worksheet Key 1.6: Geometric Sequences

  3. Geometric Sequences section 12.3 1.6: Geometric Sequences

  4. Definitions • A geometric sequence is a list of numbers that have a COMMON RATIO (r) meaning you DIVIDED by the same number. • To find r = DIVIDE THE SECOND NUMBER TO THE FIRST NUMBER 1.6: Geometric Sequences

  5. Is it Geometric? Determine whether these sequences are geometric. If it is, determine the ratio. • {1, 3, 9, 27, 81...} • {180, 90, 0, –90, –180...} • {5, 1, 0.2, 0.04, 0.008...} • {–50, –32, –18, –8, –2...} YES r = 3 Arithmetic YES r = 0.2 NO 1.6: Geometric Sequences

  6. Formulas Explicit Formula: Recursive Formula: 1.6: Geometric Sequences

  7. Steps • Find the common ratio • Use the formula and determine the rule • Evaluate for the missing variable • Check 1.6: Geometric Sequences

  8. Example 1 Find the 6th term and explicit rule for given, a1 = 5, r = 2 Step 1: Find the common ratio. 1.6: Geometric Sequences

  9. Example 1 Find the 6th term and explicit rule for given, a1 = 5, r = 2 Step 2: Use the formula and determine the term 1.6: Geometric Sequences

  10. Example 1 Find the 6th term and explicit rule for given, a1 = 5, r = 2 Step 3: Evaluate for the missing variable 1.6: Geometric Sequences

  11. Example 1 Find the 6th term and explicit rule for given, a1 = 5, r = 2 Step 4: Check 1.6: Geometric Sequences

  12. Example 2 Find the 10th term and explicit rule for the following geometric sequence of 16, –8, 4, –2,… 1.6: Geometric Sequences

  13. Your Turn Find the 5th term and explicit rule for given a1 = 1/6, a2 = –1/18. Leave answers in fraction form. 1.6: Geometric Sequences

  14. Example 3 Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 1: Find the common ratio. 1.6: Geometric Sequences

  15. Example 3 Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 2: Find a1. Consider both the positive and negative values for r. or 1.6: Geometric Sequences

  16. Example 3 Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 3: Write the rule and evaluate for a8. Consider both the positive and negative values for r. or 1.6: Geometric Sequences

  17. Example 4 Find the 8th term of the geometric sequence with a2 = 80 and a4 = 1280 1.6: Geometric Sequences

  18. Your Turn Find the 8th term of the geometric sequence and its implicit formula with a3 = 108 and a5 = 3888 1.6: Geometric Sequences

  19. Assignment Geometric Sequences Worksheet 1.6: Geometric Sequences

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