Geometric Mean

If three positive numbers $a, G, b$ are in G.P. (in that order), then $G$ is called the geometric mean (G.M.) of $a$ and $b$. Being in G.P. means the common ratio is the same on both sides: $\dfrac{G}{a}=\dfrac{b}{G}$. Cross-multiplying:

(taking the positive square root, since $a$ and $b$ are positive and the mean of two positive numbers should itself be positive). Geometrically, $G$ is the side of a square whose area equals that of the $a\times b$ rectangle — hence the name.

Inserting several geometric means. More generally, $n$ positive numbers $G_1, G_2, \ldots, G_n$ are called $n$ geometric means between $a$ and $b$ (with $a,b>0$) if $a, G_1, G_2, \ldots, G_n, b$ together form a G.P. This combined list has $n+2$ terms, with $a$ first and $b$ last. Using the general-term formula of Section 4 with this G.P.'s own common ratio $r$:

(the positive real $(n+1)$th root, so that every inserted mean is positive). Once $r$ is known, each inserted mean is found by repeatedly multiplying $a$ by $r$:

Product of the $n$ inserted means. Pairing the $k$th mean from the start with the $k$th mean from the end, exactly as was done for the arithmetic means in Section 3:

$G_k \cdot G_{n+1-k} = \big(ar^k\big)\big(ar^{n+1-k}\big) = a^2 r^{n+1} = a^2\cdot\frac{b}{a} = ab = G^2,$ …