CDS Mathematics · Sequence and Series

The Three Means

For two positive numbers, AM = (a + b)/2, GM = √(ab) and HM = 2ab/(a + b), with AM ≥ GM ≥ HM and GM² = AM × HM.

All Sequence and Series notes CDS Mathematics strategy

Why this matters

Nine PYQs, none HARD. Translate each mean into the sum or the product of the two numbers: AM gives the sum, GM the product, HM = 2 × product ÷ sum. Then the numbers are the roots of t² − (sum)t + (product) = 0.

Concept 1 of 1: AM, GM and HM of two numbers

The AM fixes the sum, the GM fixes the product, and the HM is the product divided by the average. Knowing any two of the means, you know the sum and the product, and so the numbers.

Definition

$\text{AM} = \dfrac{a + b}{2}$ , $\text{GM} = \sqrt{ab}$ , $\text{HM} = \dfrac{2ab}{a + b}$ .
$\text{GM}^2 = \text{AM} \times \text{HM}$ .
$\text{AM} \ge \text{GM} \ge \text{HM}$ , equal only when $a = b$ ; $\text{AM} - \text{GM} = \dfrac{(\sqrt a - \sqrt b)^2}{2}$ .
The numbers are the roots of $t^2 - 2\text{AM}\,t + \text{GM}^2 = 0$ .
$x$ is the HM of $y$ and $z$ exactly when $\dfrac2x = \dfrac1y + \dfrac1z$ .

The three means

\text{GM}^2 = \text{AM} \times \text{HM}

Worked example

Two numbers have AM

13

and GM

12

. Find them.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — Elementary Mathematics · Q52Easy

Example 1 · Sequence and Series · Arithmetic, Geometric and Harmonic Means

The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?

The GM squared, not the GM, is the product

A GM of

8

means the product is

64

. Using

8

as the product gives numbers that do not have the stated AM.

Drill 8 more on am, gm and hm of two numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

AM, GM and HM of two numbers
The three means
$\text{GM}^2 = \text{AM} \times \text{HM}$

Watch out for (1)

The GM squared, not the GM, is the product
→ AM, GM and HM of two numbers

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