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:

Geometric Mean=a×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:

  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:

Geometric Mean=3×27=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.