# Arithmetic Mean

#### Definition:

In sequences and series, the arithmetic mean refers to the average of two numbers in a sequence, especially in an arithmetic progression (AP).

#### Arithmetic Mean between Two Terms in an AP:

For an arithmetic progression with terms $a$ and $b$, the arithmetic mean of $a$ and $b$ is calculated as:

Where:

• $a$ and $b$ are consecutive terms in the AP.
• The arithmetic mean is the term exactly in the middle of $a$ and $b$.

#### Properties and Usage:

1. Midpoint in an AP:

• The arithmetic mean represents the midpoint between two terms in an AP.
2. Formula within an AP:

• Helps in finding an intermediate term between two given terms.

#### Example:

Consider an AP with terms $5$ and $15$.

To find the arithmetic mean:

So, the arithmetic mean between $5$ and $15$ in this AP is $10$.

#### Relationship to Arithmetic Progression:

• In an arithmetic progression, the arithmetic mean between any two consecutive terms is constant.
• The arithmetic mean can also be viewed as the term that divides any two consecutive terms into equal parts.