Geometric Mean
Definition:
In sequences and series, the geometric mean represents the middle term in a proportional sequence, especially in a geometric progression (GP).
Geometric Mean between Two Terms in a GP:
For a geometric progression with terms $a$ and $b$, the geometric mean of $a$ and $b$ is given by:
$\text{GeometricMean}=\sqrt{a\times b}$
Where:
 $a$ and $b$ are consecutive terms in the GP.
 The geometric mean is the square root of the product of $a$ and $b$.
Properties and Usage:

Middle Term in a GP:
 The geometric mean represents the middle term in a proportion sequence.

Formula within a GP:
 Helps in finding an intermediate term between two given terms.
Example:
Consider a GP with terms $3$ and $27$.
To find the geometric mean:
$\text{GeometricMean}=\sqrt{3\times 27}=\sqrt{81}=9$
So, the geometric mean between $3$ and $27$ in this GP is $9$.
Relationship to Geometric Progression:
 In a geometric progression, the geometric mean between any two consecutive terms is constant and equals the square root of their product.
 The geometric mean can also be seen as the term that lies in the geometric middle between two consecutive terms.