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Understanding Geometric Mean Sequences in Mathematics

Learn about geometric mean sequences, arithmetic sequences, and find missing terms in sequences with common factors. Discover how to find the geometric mean between numbers. Try solving geometric mean examples with provided solutions.

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Understanding Geometric Mean Sequences in Mathematics

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  1. Learning Target 3 (9.3) Geometric Mean Lesson 7-1: Geometric Mean

  2. Sequences Arithmetic Sequence: Is a pattern of numbers where any term (number in the sequence) is determined by adding or subtracting the previous term by a constant called the common difference. 17 20 23 Example: 2, 5, 8, 11, 14, ____, ____, ____ Common difference = 3 Geometric Sequence: Is a pattern of numbers where any term (number in the sequence) is determined by multiplying the previous term by a common factor. Example: 2, 6, 18, 54, 162, _____, _____, ____ 486 1458 4374 Common Factor = 3 Lesson 7-1: Geometric Mean

  3. Examples 1. Starting with the number 1 and using a factor of 4, create 5 terms of a geometric sequence. 1 , 4 , 16 , 64 , 256 2. Starting with the number 2 and using a factor of 5, create 5 terms of a geometric sequence. 2 , 10 , 50 , 250 , 1250 3. Starting with the number 5 and using a factor of 3, create 5 terms of a geometric sequence. 5 , 15 , 45 , 135 , 405 4. In the geometric sequence 2, ____, 72, 432, .Find the missing term. 12 12 5. In the geometric sequence 6, ____, 24,... Find the missing term. Lesson 7-1: Geometric Mean

  4. Geometric Mean A term between two terms of a geometric sequence is the geometric mean of the two terms. Example: In the geometric sequence 4, 20, 100, ….(with a factor of 5), 20 is the geometric mean of 4 and 100. Try It: Find the geometric mean of 3 and 300. 3 , ___ , 300 30 Lesson 7-1: Geometric Mean

  5. Geometric Mean : Fact Consecutive terms of a geometric sequence are proportional. Example: Consider the geometric sequence with a common factor 10. 4 , 40 , 400 (4)(400) = (40)(40) cross-products are equal 1600 = 1600 Lesson 7-1: Geometric Mean

  6. Therefore ……….. To find the geometric mean between 7 and 28 ... label the missing term x 7 , ___ , 28 write a proportion cross multiply solve Lesson 7-1: Geometric Mean

  7. The geometric mean between two numbers a and b is the positive number x where . Therefore x = . Try It: Find the geometric mean of . . . Answer = 20 1. 10 and 40 2. 1 and 36 Answer = 6 3. 10 and 20 Answer = 14.14 4. 5 and 6 Answer = 5.48 Answer = 9.94 5. 8.1 and 12.2 Lesson 7-1: Geometric Mean

  8. How does this relate to geometry? Lesson 7-1: Geometric Mean

  9. The " W " Pattern Lesson 7-1: Geometric Mean

  10. The Geometric Means Recall the three geometric means that you discovered from your Sketchpad activity. BUT FIRST . . . Lesson 7-1: Geometric Mean

  11. Re-label the Sides (as lengths) Lesson 7-1: Geometric Mean

  12. Geometric Mean #1 What is the proportion that uses f? f is the geometric mean of d and e. Lesson 7-1: Geometric Mean

  13. Geometric Mean #2 What is the proportion that uses b? b is the geometric mean of e and c. Lesson 7-1: Geometric Mean

  14. Geometric Mean #3 What is the proportion that uses a? a is the geometric mean of d and c. Lesson 7-1: Geometric Mean

  15. Put them all together Lesson 7-1: Geometric Mean

  16. The “W” Pattern W Lesson 7-1: Geometric Mean

  17. Try it ! Given: d = 4 and e = 10 Find: a = ___ b = ___ c = ___ f = ___ Lesson 7-1: Geometric Mean

  18. Solution: Proportions Answers Lesson 7-1: Geometric Mean

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