Geometric Mean
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The document discusses the concept of geometric means, specifically how to find the positive and negative geometric means between two numbers. It includes several examples demonstrating the insertion of geometric means between various pairs of numbers and provides formulas for calculation. References to educational resources for Grade 10 mathematics are also included.
Geometric Mean
- 1. Grade 10 – Mathematics Quarter I GEOMETRIC MEAN
- 2. FINDING THE GEOMETRIC MEAN The geometric mean between any two positive numbers 𝒂 and 𝒃 is the square root of their product. 𝒂𝒃 positive geometric mean of 𝒂 and 𝒃 - 𝒂𝒃 negative geometric mean of 𝒂 and 𝒃
- 3. Find the positive geometric mean of the following: 4 and 25 𝐆𝐌 = 𝟒 ∙ 𝟐𝟓 = 𝟏𝟎𝟎 = 𝟏𝟎 -3 and -12 𝐆𝐌 = (−𝟑)(−𝟏𝟐) = 𝟑𝟔 = 𝟔
- 4. Insert 3 geometric means between 5 and 1280. 5, ____, ____, _____, 1280 = 𝟓 ∙ 𝟏𝟐𝟖𝟎 = 𝟔𝟒𝟎𝟎 = 𝟖𝟎 𝟖𝟎 = 𝟓 ∙ 𝟖𝟎 = 𝟒𝟎𝟎 = 𝟐𝟎 𝟐𝟎 = 𝟖𝟎 ∙ 𝟏𝟐𝟖𝟎 = 𝟏𝟎𝟐𝟒𝟎𝟎 = 𝟑𝟐𝟎 𝟑𝟐𝟎
- 5. Insert 3 geometric means between 16 and 81. 16, ____, ____, ____, 81 = 𝟏𝟔 ∙ 𝟖𝟏 = 𝟏𝟐𝟗𝟔 = 𝟑𝟔 𝟑𝟔 = 𝟏𝟔 ∙ 𝟑𝟔 = 𝟓𝟕𝟔 = 𝟐𝟒 𝟐𝟒 = 𝟑𝟔 ∙ 𝟖𝟏 = 𝟐𝟗𝟏𝟔 = 𝟓𝟒 𝟓𝟒
- 6. Insert 3 geometric means between 256 and 1. 𝑎 𝑛 = 𝑎1 𝑟 𝑛−1 𝑎1 = 256 𝑎5 = 256𝑟4 Given: 1= 256𝑟4 𝑟4 = 1 256 = ± 4 1 256 = ± 1 4 𝑟 = 1 4 256, ____, ____, __, 1 𝟐𝟓𝟔 ∙ 𝟏 𝟒 𝟔𝟒 64∙ 𝟏 𝟒 𝟏𝟔 𝑟 = − 1 4 256, ____, ___, ___, 1 𝟐𝟓𝟔 ∙ − 𝟏 𝟒 −𝟔𝟒 -64∙ − 𝟏 𝟒 𝟏𝟔 16∙ 𝟏 𝟒 𝟒 16∙ − 𝟏 𝟒 -𝟒 𝑎5 = 1
- 7. Insert 2 geometric means between -32 and 4. 𝑎 𝑛 = 𝑎1 𝑟 𝑛−1 𝑎1 = −32 𝑎4 = −32𝑟3 Given: 4 = −32𝑟3 𝑟3 = − 4 32 = − 3 1 8 𝑟 = − 1 2 −32, ____, ____, 4 -32∙ − 𝟏 𝟐 𝟏𝟔 16∙ − 𝟏 𝟐 −𝟖 =− 1 8 𝑎4 = 4
- 8. If 2 and 3 are two geometric means between 𝑚 and 𝑛, find the values of 𝑚 and 𝑛. m, 2, 3, n 𝒓 = 𝟑 𝟐 𝑚 = 2 ÷ 3 2 = 2 ∙ 2 3 𝒎 = 𝟒 𝟑 𝑛 = 3 ∙ 3 2 = 9 2 𝒏 = 𝟗 𝟐
- 9. REFERENCES: ❖ Nivera, G.C. (2015). Grade 10 Mathematics Pattern and Practicalities. Don Bosco Press, Inc. Makati City, Philippines. ❖ Mathematics Grade 10 Learner’s Module. Department of Education. Pasig City, Philippines.