Arithmetic, Geometric and Harmonic Mean

The familiar idea of an "average" comes in three flavours here: arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM).

Definition 5.3 (AM). For $n$ numbers $a_1,\ldots,a_n$ (need not be in AP, need not be distinct, need not be positive), the arithmetic mean is

Definition 5.4 (GM). For $n$ non-negative numbers $a_1,\ldots,a_n$, the geometric mean is

(replacing "add, then divide by $n$" with "multiply, then take the $n^{th}$ root"). E.g. the GM of $4,6,9$ is $\sqrt[3]{216}=6$, while their AM is $\tfrac{19}3=6\tfrac13$ — the AM is larger, and this is no accident.

Theorem 5.2 ($AM\ge GM$, two numbers). For non-negative $a,b$: $AM=\tfrac{a+b}2$, $GM=\sqrt{ab}$. Since $(a+b)^2-4ab=(a-b)^2\ge0$, we get $(a+b)^2\ge4ab$, i.e. $a+b\ge2\sqrt{ab}$, i.e. $\tfrac{a+b}2\ge\sqrt{ab}$. Equality holds iff $(a-b)^2=0$, i.e. $a=b$.

Geometrical proof. Draw segment $AB=a+b$ with midpoint $M$ (so $AM_{\text{seg}}=MB=\tfrac{a+b}2$, the radius of the semicircle on $AB$), mark $D$ on $AB$ with $AD=a,DB=b$, and erect the perpendicular at $D$ meeting the semicircle at $C$. Similar triangles $\triangle ACD,\triangle CBD$ give $CD^2=AD\cdot DB=ab$, so $CD=\sqrt{ab}$; since any half-chord is at most the radius, $CD\le CM=\tfrac{a+b}2$, i.e. $GM\le AM$, with equality iff $D=M$ (i.e. $a=b$).

Result 5.1 / Result 5.2. In an AP, every term (after the first) is the AM of its neighbours; in a GP, every term is the GM of its neighbours — both proved directly from the $n^{th}$-term formulas.

Harmonic mean. For positive numbers $h_1,\ldots,h_n$, the HM is the reciprocal of the AM of the reciprocals:

Theorem 5.3 ($GM\ge HM$). $GM-HM=\sqrt{ab}-\dfrac{2ab}{a+b}=\dfrac{\sqrt{ab}(\sqrt a-\sqrt b)^2}{a+b}\ge0$, with equality iff $a=b$.

Combining Theorems 5.2 and 5.3: $AM\ge GM\ge HM$ always (two positive numbers), equality throughout iff the numbers are equal.

Result 5.3. For any two positive numbers, $AM\times HM=\left(\tfrac{a+b}2\right)\left(\tfrac{2ab}{a+b}\right)=ab=(\sqrt{ab})^2=GM^2$ — so $AM,GM,HM$ are themselves in GP.

Standing facts: if $b$ is the AM of $a,c$ then $a,b,c$ is an AP; if $b$ is the GM of $a,c$ then $a,b,c$ is a GP; if $b$ is the HM of $a,c$ then $a,b,c$ is an HP. …