If A, G, and H denote the arithmetic mean, geometric mean, and harmonic mean (for two positive numbers), which identity holds among them?

Introduction / Context:
For two positive numbers, the classical relationships among arithmetic mean (A), geometric mean (G), and harmonic mean (H) include a useful identity that links them exactly, not just via inequalities.

Given Data / Assumptions:

• Two positive numbers x and y.
• Definitions: A = (x + y)/2, G = sqrt(xy), H = 2xy/(x + y).

Concept / Approach:

• Compute A*H using the definitions, and compare to G^2.

Step-by-Step Solution:

Verification / Alternative check:
Pick x = 4, y = 9: A = 6.5, G = 6, H = 2*36/13 ≈ 5.538; A*H ≈ 36 = G^2; identity holds.

Why Other Options Are Wrong:

• A × H = G and the ratio forms do not simplify to xy consistently.
• None of these: Not applicable; A × H = G^2 is exact for two numbers.

Common Pitfalls:

• Confusing the AM–GM–HM inequalities with this identity.
• Applying the identity to more than two numbers, where it does not generally hold.

Final Answer:
A × H = G^2