Relationship between A.M and G.M

A.M stands for Arithmetic Mean and G.M stands for Geometric Mean. Here we will be learning about the relationship between Arithmetic Mean and Geometric Mean.

Arithmetic Mean

The arithmetic mean (AM) is a number obtained by dividing the sum of a set’s values by the total number of values in the set.

If $a_{1},a_{2},a_{3}....a_{n}$ is a number of group of values or the Arithmetic Progression, then

$AM={\frac {(a_{1}+a_{2}+a_{3}....+a_{n})}{n}}$

Geometric Mean

The geometric mean (GM) is a number that is obtained by multiplying the series together and then taking the $n^{th}$ root of the result where $n$ is the number of terms.

$GM={\sqrt[{n}]{a_{1}\times a_{2}\times a_{3}\times ....\times a_{n}}}$

Relationship between Arithmetic Mean and Geometric Mean

Let $A$ and $G$ be A.M. and G.M. of two given positive real numbers $a$ and $A$, respectively.

Then $A={\frac {a+b}{2}}$ and $G={\sqrt {ab}}$

$A-G={\frac {a+b}{2}}-{\sqrt {ab}}={\frac {a+b-2{\sqrt {ab}}}{2}}={\frac {({\sqrt {a}}-{\sqrt {b}})^{2}}{2}}$