Harmonic Progression and AM-GM-HM Inequality | NIMCET

Quick answer: A sequence is a harmonic progression (HP) if the reciprocals of its terms form an AP. The harmonic mean of a and b is $H = \frac{2ab}{a + b}$. For positive a and b, $A \ge G \ge H$, where $A = \frac{a + b}{2}$ and $G = \sqrt{ab}$, with equality only when $a = b$, and $G^2 = AH$. An arithmetico-geometric series multiplies an AP by a GP term by term.

This brings together arithmetic and geometric progressions.

Harmonic progression

If $a_1, a_2, a_3, \ldots$ is an HP, then $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots$ is an AP with common difference d, and

$a_n = \frac{1}{\frac{1}{a_1} + (n - 1)d}$

There is no simple formula for the sum of an HP.

• a, b, c are in HP if and only if $b = \frac{2ac}{a + c}$.
• Equivalently, $\frac{a - b}{b - c} = \frac{a}{c}$.

The three means

Relations between the means

For positive numbers:

$A \ge G \ge H$

with equality only when all numbers are equal. For two numbers, $G^2 = AH$, so A, G, H are in GP.

Recovering the numbers: a and b are the roots of $x^2 - 2Ax + G^2 = 0$.

AM-GM for extreme values

For positive terms with a fixed product, the sum is smallest when the terms are equal; with a fixed sum, the product is largest when the terms are equal.

Arithmetico-geometric series

The general term is $[a + (n - 1)d]r^{n - 1}$. To sum it, write S, multiply by r, subtract, and sum the resulting GP.

For $\lvert r\rvert < 1$:

$S_\infty = \frac{a}{1 - r} + \frac{dr}{(1 - r)^2}$

Worked examples

Example 1: nth term of an HP

Find the 10th term of the HP $\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \ldots$.

The reciprocals 2, 5, 8, … have $d = 3$, so the 10th reciprocal is $2 + 27 = 29$ and the term is $\frac{1}{29}$.

Example 2: the three means

For 4 and 16: $A = 10$, $G = 8$, $H = \frac{128}{20} = 6.4$.

Check: $AH = 64 = G^2$ ✓.

Example 3: minimum by AM-GM

Find the minimum of $4x + \frac{9}{x}$ for $x > 0$.

$4x + \frac{9}{x} \ge 2\sqrt{36} = 12$, with equality at $4x = \frac{9}{x}$, that is, $x = \frac{3}{2}$.

Example 4: AGP sum

Find $1 + 2x + 3x^2 + 4x^3 + \ldots$ for $\lvert x\rvert < 1$.

Here $a = 1$, $d = 1$, $r = x$:

$S_\infty = \frac{1}{1 - x} + \frac{x}{(1 - x)^2} = \frac{1}{(1 - x)^2}$

Example 5: numbers from the means

The AM and GM of two numbers are 5 and 4. Find the numbers.

$x^2 - 10x + 16 = 0$, so the numbers are 2 and 8.

Summary table

Exam tips

• Turn every HP question into an AP question by taking reciprocals.
• AM-GM needs positive terms; check before applying it.
• Remember $1 + 2x + 3x^2 + \ldots = \frac{1}{(1 - x)^2}$; it appears in probability too.

Next, read matrices: types, multiplication and transpose, or browse the NIMCET Algebra chapter.

Common mistakes

• Trying to use a sum formula for an HP.
• Applying AM-GM to negative numbers.
• Writing $H = \frac{a + b}{2ab}$, which is the reciprocal of the correct value.
• Forgetting that equality in AM ≥ GM needs equal terms, which may be impossible under the constraints.

Frequently Asked Questions

What is a harmonic progression?

A sequence whose reciprocals form an arithmetic progression.

What is the harmonic mean of two numbers?

H = 2ab/(a + b).

What is the relation between AM, GM and HM?

For positive numbers, AM ≥ GM ≥ HM. For two numbers, GM² = AM × HM.

What is the minimum value of x + 1/x for x > 0?

It is 2, reached at x = 1.

What is the sum of 1 + 2x + 3x² + … for |x| < 1?

It equals 1/(1 − x)².