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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.