Harmonic progression — definition, HM: Question bank

Recall Notation reminder (what the letters mean)

$H$ and HM ::: the same quantity — the harmonic mean $H = HM (a, b) = \frac{2 ab}{a + b}$ ; we write $H$ for short.
AM ::: arithmetic mean $\frac{a + b}{2}$ (the everyday average).
GM ::: geometric mean $ab$ (the "multiply-then-root" average, see Geometric Mean).
$\frac{1}{a}$ ::: the reciprocal of $a$ — flip numerator and denominator.

Two pictures anchor the whole page. Figure 1 shows the "flip" mapping that turns an HP into an AP. Figure 2 shows why $AM \geq GM \geq HM$ — the three means literally stack in that order.

Figure — Harmonic progression — definition, HM

Figure 1 — The blue dots (top) are an HP: $6, 3, 2$ . Each gray vertical arrow is the reciprocal-flip map, sending term $a$ down to $\frac{1}{a}$ . The orange dots (bottom) are the flipped values $\frac{1}{6}, \frac{1}{3}, \frac{1}{2}$ ; the green double-arrows show their gaps are both $+ \frac{1}{6}$ — a constant difference, so the flip has revealed a hidden AP.

Figure 2 — On a single number line the endpoints $a = 2$ and $b = 8$ are gray. The three means land in a fixed order: red HM $= 3.20$ , green GM $= 4.00$ , blue AM $= 5.00$ . Read left-to-right to see $HM \leq GM \leq AM$ ; all three would merge into one dot only if $a = b$ .

True or false — justify

A sequence is in HP whenever its terms keep decreasing.

False — decreasing is not enough; the reciprocals must have a constant difference. Many decreasing lists (like

8, 4, 1

) are not HP.

Every AP with no zero terms becomes an HP when you flip each term.

True — flipping an AP's terms gives an HP by definition, provided no term is

0

so every reciprocal exists.

The constant sequence

5, 5, 5, \dots

is an HP.

True — its reciprocals

\frac{1}{5}, \frac{1}{5}, \frac{1}{5}

form an AP with common difference

0

, which satisfies the definition.

a, b, c

are in HP then

a, b, c

are also in AP.

False in general — that would force

\frac{1}{a}, \frac{1}{b}, \frac{1}{c}

and

a, b, c

to both be APs, which only happens when all three are equal.

The HM of two positive numbers always lies between them.

True —

HM (a, b)

sits strictly between

min

and

max

(equal to both only when

a = b

), just like any genuine average.

HM (a, b) = \frac{1}{2} (\frac{1}{a} + \frac{1}{b})

False — that expression is

\frac{1}{H}

(the average of reciprocals); the HM is its reciprocal

H = \frac{2 ab}{a + b}

For positive numbers,

AM \geq GM \geq HM

with equality only when the numbers are equal.

True — the chain collapses to a single value exactly when

a = b

; otherwise every inequality is strict.

An HP can contain a term equal to

0

False — a

0

term has no reciprocal, so the reciprocal AP can't exist; HP forbids zero terms entirely.

A single HP can contain both positive and negative terms.

True — the reciprocal AP may start positive and later become negative; as long as it never lands on

0

, the flipped-back terms simply switch sign at that point (unpacked below in Edge cases).

Spot the error

4, 6, 9

is in HP because it grows by a nice ratio."

Growing by a ratio makes it a Geometric Progression, not HP; you must flip and check the reciprocals for a constant difference, which

\frac{1}{4}, \frac{1}{6}, \frac{1}{9}

do not have.

"The average speed of going at

u

then

v

\frac{u + v}{2}

That AM is only correct for equal times; for equal distances the times (reciprocals of speed) add, giving the HM

\frac{2 uv}{u + v}

instead — see Average speed and rates.

HM (a, b) = \frac{a + b}{2 ab}

The numerator and denominator are swapped:

\frac{a + b}{2 ab}

equals

\frac{1}{H}

, so the HM is its reciprocal

\frac{2 ab}{a + b}

"The middle term of any HP

a, H, b

is the ordinary average

\frac{a + b}{2}

The reciprocal is the average (that's what "AP in the middle" means); flipping back gives HM, which is smaller than

\frac{a + b}{2}

for positive unequal

a, b

"Sum of an HP has a closed form like

\frac{n}{2} (a + l)

No — there is no simple closed form for

\sum \frac{1}{A + ( n - 1 ) D}

; that AP-style sum formula does not transfer through the reciprocal flip.

"Since

GM^{2} = AM \cdot HM

, the GM is the average of AM and HM."

The GM is their geometric mean, not arithmetic —

GM = AM \cdot HM

, so it equals

\frac{1}{2} (AM + HM)

only when all are equal.

"Two resistors of

3 Ω

and

6 Ω

in parallel give the AM,

4.5 Ω

Parallel resistors add as reciprocals, so the combined value relates to the HM, not the AM — see Resistors in parallel; the result (

2 Ω

) is less than either resistor.

Why questions

Why is HP defined through reciprocals instead of getting its own fresh theory?

Because flipping the terms turns HP into an Arithmetic Progression, letting us reuse all AP tools (nth term, means) instead of reinventing them.

Why does average speed over equal distances use HM and not Arithmetic Mean?

Equal distances make the times (each

= \frac{d}{speed}

) add up, and times are reciprocals of speed, so the speeds combine harmonically.

Why must we "flip back at the very end" when finding an HP's nth term?

The AP nth-term formula gives the reciprocal

\frac{1}{a _{n}}

; the term we actually want is

a_{n}

, so a final reciprocal undoes the disguise.

Why is the HM the smallest of the three means for positive unequal numbers?

Reciprocating is order-reversing, so the largest reciprocals dominate the average, pulling the flipped-back result down below GM and AM (this is exactly Figure 2).

Why does

\frac{1}{H} = \frac{1}{2} (\frac{1}{a} + \frac{1}{b})

come from "the middle of an AP"?

Equal gaps mean

\frac{1}{H} - \frac{1}{a} = \frac{1}{b} - \frac{1}{H}

; solving shows the middle reciprocal is the average of the end reciprocals.

Why can't we just check "constant ratio" to test for HP?

Constant ratio is the test for a Geometric Progression; HP requires the reciprocals to have a constant difference, a completely different condition.

Why does the lens formula feel harmonic?

Because

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}

adds reciprocals of distances, the same reciprocal-adding structure behind HM — see Lens formula.

Edge cases

What is the HM of a number with itself,

HM (a, a)

It equals

a

— the chain

AM = GM = HM = a

collapses to a single value whenever the numbers are equal.

What is

HM (a, - a)

when the two numbers are opposites?

Undefined — then

a + b = 0

, so

H = \frac{2 ab}{a + b}

divides by zero; opposite pairs have no harmonic mean, which is why this case must be forbidden, not "answered".

Can an HP be strictly increasing?

Yes — if the reciprocal AP is strictly decreasing (negative common difference), the flipped terms strictly increase, e.g. reciprocals

\frac{1}{6}, \frac{1}{5}, \frac{1}{4}

give HP

6, 5, 4

Show a concrete HP that mixes signs.

Take a reciprocal AP that steps down past (but never onto) zero:

\frac{3}{2}, \frac{1}{2}, - \frac{1}{2}, - \frac{3}{2}

(constant difference

- 1

, never

0

); flipping gives HP

\frac{2}{3}, 2, - 2, - \frac{2}{3}

, positive then negative — legal because no reciprocal is

0

What happens to HM

(a, b)

b \to 0^{+}

with

a

fixed positive?

H = \frac{2 ab}{a + b} \to 0

— a tiny value drags the harmonic mean toward

0

far more strongly than it would drag the AM.

What happens to average speed if the return speed

v \to \infty

\overset{s}{ˉ} = \frac{2 uv}{u + v} \to 2 u

— the round-trip average caps at twice the slow speed, never higher, because the slow leg dominates the total time.

Is a two-term "sequence" like

a, b

enough to call it an HP?

Any two nonzero numbers trivially form an HP (and an AP), since two reciprocals always have a well-defined single difference; HP structure only becomes meaningful from three terms on.

Does

GM^{2} = AM \cdot HM

still hold if one number is negative?

Not reliably — the identity is derived and stated for positive

a, b

; with a negative number the GM

ab

may not even be real.

Recall One-line summary of every trap

Flip → it's an AP → do AP work → flip back. Constant difference of reciprocals (not ratio), no zero terms, no opposite pairs, and $\frac{1}{H}$ is the average — the HM is its reciprocal.

Connections

Arithmetic Progression — the disguise HP hides.
Geometric Progression — the ratio test people confuse with HP; also the middle mean.
Arithmetic Mean / Geometric Mean — partners of HM in the AM–GM–HM chain.
Average speed and rates, Resistors in parallel, Lens formula — physical homes of reciprocal-adding.
3.3.04 Harmonic progression — definition, HM (Hinglish) — the same ideas in Hinglish.