If A, G and H are A.M, G.M and H.M. of any two given...

If A, G and H are A.M, G.M and H.M. of any two given positive numbers, then find the relation between A, G and $\mathrm{H}$.
(1) $\mathrm{A}^{2}=\mathrm{GH}$
(2) $\mathrm{G}^{2}=\mathrm{AH}$
(3) $\mathrm{H}^{2}=\mathrm{AG}$
(4) $\mathrm{G}^{3}=\mathrm{A}^{2} \mathrm{H}$

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Key Concepts

Relations between Means

For any two positive numbers, there exists a specific relationship between their arithmetic, geometric, and harmonic means. In particular, the product of the arithmetic and harmonic means equals the square of the geometric mean. This relation, G² = A × H, encapsulates how these different means are interlinked and highlights an intrinsic balance between them.

Harmonic Mean

The harmonic mean of two numbers is defined as twice the product of the two numbers divided by their sum. It is especially relevant when dealing with rates or ratios and provides a way to average quantities that are defined in relation to some unit measure, such as speed or density.

Geometric Mean

The geometric mean of two positive numbers is the square root of their product. This mean is particularly useful in contexts where the numbers are multiplied together or when dealing with rates of growth, and it is a more appropriate measure than the arithmetic mean in such cases.

Arithmetic Mean

The arithmetic mean of two numbers is calculated by summing the numbers and then dividing by two. It is a measure of central tendency that provides a simple average and is foundational in many areas of mathematics and statistics.

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