Tutoring high school math, you’re sometimes asked about geometric mean.

Imagine a number, call it n. You multiply it by a number, r, to get nr. Then you multiply it by r again, to get nr².

n   nr   nr²

In the sequence above, the geometric mean is nr, the middle number.

Consider the numerical example

5   20   80

Unlike the arithmetic mean, the geometric mean is not equidistant from the earlier and later numbers. It’s one multiple by r from the first, and one division by r from the last.

In most problems, finding that multiplying number r is the critical idea.

Example: Find the geometric mean between 14 and 42.

Solution: We imagine the sequence

14   ?   42

At the same time, we can imagine that sequence as

n   nr   nr²

From the first two representations, we put together

14   14r   14r²=42

From

14r²=42

we divide both sides by 14, to get

r²=42/14=3

Finally, square rooting both sides, we arrive at

r=±√3

Apparently, our geometric mean, known before as 14r, turns out to have two possible values: 14√3, or else -14√3.

I’ll be continuing this topic in future posts. HTH:)

Source:

Travers, Kenneth et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited,    1977.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.