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Warm-Up

Warm-Up. Geometric Mean is where two constant terms of a geometric sequence with exactly one term in between them. Definition. Geometric Mean is where two constant terms of a geometric sequence with exactly one term in between them. Geometric Sequences. section 12.3. Definition.

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Warm-Up

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  1. Warm-Up • Geometric Meanis where two constant terms of a geometric sequence with exactly one term in between them. 12.3 Geometric Sequences

  2. Definition • Geometric Meanis where two constant terms of a geometric sequence with exactly one term in between them. 12.3 Geometric Sequences

  3. Geometric Sequences section 12.3 12.3 Geometric Sequences

  4. Definition • Geometric Meanis where two constant terms of a geometric sequence with exactly one term in between them. 12.3 Geometric Sequences

  5. Example 1 Determine the Geometric Mean for 16 and 25. 12.3 Geometric Sequences

  6. Your Turn Determine the Geometric Mean for 1/2 and 1/32. 12.3 Geometric Sequences

  7. Definitions • Geometric Sequenceis where consecutive terms have a common ratio • Ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another 12.3 Geometric Sequences

  8. Is it Geometric? • Determine whether these sequences are geometric. If it is, determine the ratio. • 1, 3, 9, 27, 81... • 180, 90, 60, 15, 3.75 ... • 5, 1, 0.2, 0.04, 0.008 ... • –50, –32, –18, –8, –2 ... YES r = 3 NO YES r = 0.2 NO 12.3 Geometric Sequences

  9. Steps • Find the common ratio (can make a t-chart) • Use the formula and determine the rule • Evaluate for the missing variable • Check using graphing calculator 12.3 Geometric Sequences

  10. Example 1 For this geometric sequence, determine the first four terms of Step 1: Find the common ratio. 12.3 Geometric Sequences

  11. Example 1 For this geometric sequence, determine the first four terms of Step 2: Use the formula and determine the rule 12.3 Geometric Sequences

  12. Example 1 For this geometric sequence, determine the first four terms of Step 3: Evaluate for the missing variable 12.3 Geometric Sequences

  13. Example 1 For this geometric sequence, determine the first four terms of Step 3: Evaluate for the missing variable 12.3 Geometric Sequences

  14. Example 1 For this geometric sequence, determine the first four terms of Step 4: Check using graphing Calculator 12.3 Geometric Sequences

  15. Example 2 For this geometric sequence, determine the first four terms of 12.3 Geometric Sequences

  16. Your Turn For this geometric sequence, determine the first four terms of 12.3 Geometric Sequences

  17. Example 3 Given the first term and the common ratio of a geometric sequence, determine the first five terms and the explicit formula, an = a1r (n-1) with a1 = 10 and r = 3. NO DECIMALS! 12.3 Geometric Sequences

  18. Your Turn Given the first term and the common ratio of a geometric sequence, determine the first five terms and the explicit formula, an = a1r (n-1) with a1 = 5 and r = 1/2. NO DECIMALS! Also, determine a11. 12.3 Geometric Sequences

  19. Example 4 Find the 7th term of this geometric sequence of 1, 3, 9, 27, 81... Step 1: Determine the Ratio 12.3 Geometric Sequences

  20. Example 4 Find the 7th term of this geometric sequence of 1, 3, 9, 27, 81... Step 2: Use the formula and determine the rule 12.3 Geometric Sequences

  21. Example 4 Find the 7th term of this geometric sequence of 1, 3, 9, 27, 81... Step 3: Evaluate for the missing variable 12.3 Geometric Sequences

  22. Example 4 Find the 7th term of this geometric sequence of 1, 3, 9, 27, 81... Step 4: Check 12.3 Geometric Sequences

  23. Example 5 Find the 7th term of this geometric sequence of 3, 12, 48, 192, .... 12.3 Geometric Sequences

  24. Your Turn Find the 8th term of this geometric sequence of 12.3 Geometric Sequences

  25. Example 6 Find the 8th term of the geometric sequence with a3 = 36 and a5 = 324. Step 1: Find the common ratio. a5=a3r(5– 3) Use the given terms. Simplify. a5 = a3r2 Substitute 324 for a5 and 36 for a3. 324=36r2 Divide both sides by 36. 9 = r2 3 = r Take the square root of both sides. 12.3 Geometric Sequences

  26. Example 6 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. an =a1r n - 1 an =a1r n - 1 General rule 36 = a1(3)3- 1 36= a1(–3)3- 1 or Use a3 =36 and r = 3. 4 = a1 4 = a1 12.3 Geometric Sequences

  27. Example 6 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. an =a1r n - 1 an =a1r n - 1 General rule or an =4(3)n - 1 an =4(–3)n - 1 Substitute a1 and r. a8=4(3)8 - 1 a8 =4(–3)8- 1 Evaluate for n = 8. a8 =8748 a8 =–8748 The 8th term is 8748 or –8748 12.3 Geometric Sequences

  28. Your Turn Find the 8th term of the geometric sequence with a2 = 80 and a4 = 1280 12.3 Geometric Sequences

  29. Assignment WKST 2 Quiz Tuesday 12.3 Geometric Sequences

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