# 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:

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:

1. Middle Term in a GP:

• The geometric mean represents the middle term in a proportion sequence.
2. 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:

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.