Geometric Mean (G.M.)

The Geometric Mean of Two Numbers

For any two positive numbers $a$ and $b$, their geometric mean (G.M.) is defined as $\sqrt{ab}$.

For example, the geometric mean of $2$ and $8$ is $\sqrt{2 \times 8} = \sqrt{16} = 4$.

What makes this number special? Notice that $2, 4, 8$ are three consecutive terms of a geometric progression (G.P.) with common ratio $2$. This is not a coincidence — it points to a deeper idea: the geometric mean of two numbers is the number that sits exactly between them in a G.P.

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Inserting Multiple Geometric Means Between Two Numbers

The idea extends naturally. Given any two positive numbers $a$ and $b$, we can insert as many numbers as we like between them so that the entire sequence — including $a$ and $b$ — forms a G.P.

Let $G_1, G_2, G_3, \dots, G_n$ be $n$ numbers inserted between $a$ and $b$ such that

is a G.P.

Here $a$ is the first term, and $b$ is the $(n+2)$-th term. If the common ratio is $r$, then

From this we get

Now each inserted geometric mean can be written explicitly:

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Worked Example: Inserting Three Numbers

Example 12. Insert three numbers between $1$ and $256$ so that the resulting sequence is a G.P.

Solution. Let the three numbers be $G_1, G_2, G_3$. We want

to be a G.P. Here $a = 1$, $b = 256$, and $n = 3$ (three inserted terms). The total number of terms is $5$, so $b$ is the $5$-th term:

Taking real roots only, $r = \pm 4$.

Case 1: $r = 4$

So the sequence is $1, 4, 16, 64, 256$.

Case 2: $r = -4$

$G_1 = 1 \cdot (-4) = -4, \quad G_2 = 1 \cdot (-4)^2 = 16, \quad G_3 = 1 \cdot (-4)^3 = -64.$ …