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:

Arithmetic Mean=a+b2

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:

Arithmetic Mean=5+152=202=10

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.