If A, G, and H denote the arithmetic mean, geometric mean, and harmonic mean (for two positive numbers), which identity holds among them?
Aptitude
Odd Man Out and Series
Difficulty: Easy
Choose an option
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AA x H = G
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BA x H = G2
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CA/H = G2
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DA/H = G
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ENone of these
Answer
Correct Answer: A x H = G2
Explanation
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:
A * H = [(x + y)/2] * [2xy/(x + y)] = xyG^2 = (sqrt(xy))^2 = xyTherefore, A * H = G^2Verification / 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