Harmonic Mean when AM Equals Square of GM

Introduction

In competitive exams, questions connecting arithmetic mean, geometric mean and harmonic mean are very common. One typical problem asks what the harmonic mean is when the arithmetic mean of two positive numbers equals the square of their geometric mean.

Understanding the standard formulas and the key relation GM² = AM × HM makes such questions quick and scoring.

Step-by-Step Solution

Let the two positive numbers be a and b.

• Arithmetic Mean (AM): AM = (a + b)/2
• Geometric Mean (GM): GM = √(ab)
• Harmonic Mean (HM): HM = 2ab/(a + b)

Given condition: “The arithmetic mean is equal to the square of the geometric mean”

(a + b)/2 = (GM)² = (√(ab))² = ab

So, (a + b)/2 = ab ⇒ a + b = 2ab

Now substitute this into the HM formula:

HM = 2ab/(a + b) = 2ab/(2ab) = 1

Hence, the harmonic mean is 1, which matches option (a).

Key Relation Used

For two positive numbers, there is a standard relation between the three means: GM² = AM × HM

In this problem, the condition AM = GM² is given. Combine it with the standard relation:

GM² = AM × HM, and AM = GM²

Substitute AM = GM² into the first equation: GM² = GM² × HM

Since GM is positive, divide by GM²: 1 = HM

This confirms HM = 1.

Explanation of Each Option

• Option (a) 1 – Correct. Both direct substitution in the HM formula and the standard identity GM² = AM × HM give HM = 1.
• Option (b) 0 – Incorrect. For positive numbers, the harmonic mean is always positive, so it cannot be 0. Also, if HM were 0, then GM² = AM × 0 = 0, contradicting that GM of positive numbers is positive.
• Option (c) ½ – Incorrect. Putting HM = 1/2 in GM² = AM × HM gives GM² = AM/2, which contradicts the given condition AM = GM².
• Option (d) Cannot say – Incorrect. Using the formulas and the relation between AM, GM and HM, the harmonic mean is uniquely determined as 1.

Why This Question is Important for Exams

• It tests direct application of AM, GM and HM formulas for two numbers.
• It checks whether the test-taker knows the key identity GM² = AM × HM.