Geometric Mean Formula

Geometric Mean Formula

Small Description: Measures of the central tendency are used in mathematics and statistics to express an overview of the values that are present in a data collection.

“Mathematics is queen of science and arithmetic is queen of mathematics.”  These were the words of Carl Gauss, the person who gave the world sequence and series arithmetic progression concept along with several other major contributions in the field of mathematics, A.P is a special type of sequence where the difference between the two consecutive terms is always the same/Constant. In arithmetic progression the next term is obtained by adding or subtracting the common difference to or from the previous term whereas when we are multiplying OR dividing the previous term with any constant. Geometric progression is defined as the sequence formed by multiplying the previous term by any non-zero constant known as the common ratio (G.P.). Working with A.P. and G.P. leads to the arithmetic mean, geometric mean, and finally the harmonic mean. In this article we shall work out with geometric mean.

Geometric Progression or Geometric Mean

Consider the following series

5, 25, 125, 625

Common ratio(r) here is 25/5 = 5

a = 5

by using the formula an = arn-1, we can easily obtain the 4th term as

a4 = 5 * 53 = 5*125 = 625

now let us try to find sum of all the terms in above series using the formula s =

s = 5(624)/4 = 5*156 = 780 

now in the above series consider the first 3 terms i.e., 5, 25, 125

it can be represented as p, q, r

now the series to be in Geometric mean

it should be in the form of 

q2 = pr or q = squar root of pr 

This is the geometric mean.

Now coming back to the above series, suppose we need to insert a number between 5 and 125 such that the series gets into G.P then we need to just make use of the derived formula to insert such number

And the number would be  = 25

Solved Example

1. Find the geometric mean (G.M) of the numbers 18 and 8.

Here x = 18 and y = 8

G = 12 

2. If 27. x and 3 are in geometric progression then determine the value of x.

Since the above numbers are in geometric progression, we can simply write them as below:

Therefore,

So, x = 9

AM (arithmetic mean) and GM (geometric mean) relationship

Consider  x and y to be  two numbers then,

AM = (x+y)/2 and GM =square root of xy

AM – GM =is always greater than 0

Therefore, we can conclude that Arithmetic Mean is always equal to or greater than Geometric Mean.