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Geometric Mean and Radicals Keystone Geometry 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. Common Factor = 3 Example: 2, 6, 18, 54, 162, _____, 486 1458 _____, 4374 ____ 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 12 72, 432… 4. Find the missing term in the geometric sequence 2, ____, 12 24,... 5. Find the missing term in the geometric sequence 6, ____, 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 , ___ 30 , 300 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 40 cross-products are equal = 40 400 (4)(400) = (40)(40) 1600 = 1600 6 Therefore ……….. To find the geometric mean between 7 and 28 ... 7 , ___ X , 28 label the missing term x 7 X = X 28 write a proportion cross multiply solve X2 = (7)(28) X2 = 196 X2 = 196 X = 14 7 The geometric mean between two numbers a and b is the positive number x where a = x . Therefore x = x b ab Try It: Find the geometric mean of . . . 1. 10 and 40 Answer = 20 2. 1 and 36 Answer = 6 3. 10 and 20 Answer = 14.14 4. 5 and 6 Answer = 5.48 5. 8.1 and 12.2 Answer = 9.94