Geometric mean of numbers

In this publication, we will consider using which formula you can find the geometric mean of numbers, and also analyze examples of tasks to demonstrate it in practice.

Calculation of the geometric mean

To calculate the geometric mean of two or more numbers, you need to multiply them, and then extract the root from the result, the degree of which is equal to their number.

Let’s say we have numbers a₁,₂, … , a_n. The geometric mean is found by the formula:

Special cases of the formula:

Example of a problem

Task 1

Find the geometric mean of the numbers 3, 6 and 12.

Decision:

Let’s use the corresponding formula for three numbers:

Task 2

The geometric mean of four numbers is 4, and three of them are also known – 2, 2 and 4. Let’s find the fourth.

Decision:

Let’s denote the number to be found by the letter x. The formula looks like this:

We place the number 4 under the root sign, preserving the equality (to do this, we raise it to the fourth power, i.e. 4⁴ = 256):

Consequently, the x = 256 : 16 = 16.