What is the ordering relationship between the harmonic, geometric, and arithmetic means for distinct positive numbers?

Conditions

• Inputs are positive real numbers
• Comparing H, G, and A means

Reasoning, step by step

1. Recall the definitions: $H = \frac{2}{1/a+1/b}$, $G = \sqrt{ab}$, $A = \frac{a+b}{2}$.
2. Observe the example values: For 40 and 60, $H=48$, $G \approx 48.99$, $A=50$.
3. Generalize the inequality chain: $H < G < A$ for $a \neq b$.
4. Note the condition for equality: $H=G=A$ iff $a=b$.

Example

In the travel example with speeds 40 and 60, the harmonic mean is 48, the geometric mean is $\sqrt{2400} \approx 48.99$, and the arithmetic mean is 50. This confirms $48 < 48.99 < 50$.

Common misconceptions

• Believing the arithmetic mean is always smaller than the others.
• Thinking the order depends on the specific values chosen rather than the type of mean.

Watch the explanation

Explore next

• Can you prove why $H \le G$ algebraically?
• Do these inequalities hold for negative numbers?