Definition of Geometric Mean:
If three quantities are in Geometric Progression then the middle one is called the geometric mean of the other two.
Let, three numbers a, G and b are in Geometric Progression then, the middle number G is called the geometric mean between two numbers a and b.
⇔ a, G, b are in Geometric Progression
⇔ \(\frac{G}{a}\) = \(\frac{b}{G}\) = common ratio.
⇔ G\(^{2}\) = ab
⇔ G = ±√ab
Solved examples on Geometric Mean
1. In the Geometric Progression {3, 9, 27}, 9 is the geometric mean of 3 and 27.
2. The geometric mean between 3 and 12 is given by G = √(3 X 12) = √36 = 6
3. The geometric mean between -3 and -27 is given by G =√(-3) X (-27) = - 9
Therefore, the geometric mean of two given quantities is any one of the two square root of their product.
When more than three quantities are in Geometric Progression then, the quantities between the two extremes are called the geometric means of the extreme quantities.
Therefore, in the Geometric Progression {4, 8, 16, 32, 64} the terms 8, 16 and 32 are the geometric means of the extreme terms 4 and 64.
Similarly, in the Geometric Progression {5, 15, 45, 135, 405, 1215, 3645} the terms 15, 45, 135, 405 and 1215 are the geometric means of the extreme terms 5 and 3645.
Notes:
When a and b are two quantities of opposite symbols then, the geometric mean between these quantities does not exist.
● Geometric Progression
11 and 12 Grade Math
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