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Geometric Mean

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Geometric Mean

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References:
Nivera, G. C. (2015), Grade 10 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

Mathematics Grade 10 Learner's Module (2015). Department of Education

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References:
Nivera, G. C. (2015), Grade 10 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.

Mathematics Grade 10 Learner's Module (2015). Department of Education

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Geometric Mean

  1. 1. Grade 10 – Mathematics Quarter I GEOMETRIC MEAN
  2. 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. 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. 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. 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. 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. 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. 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. 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.

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