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 and , the geometric mean of and is given by:
- and are consecutive terms in the GP.
- The geometric mean is the square root of the product of and .
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.
Consider a GP with terms and .
To find the geometric mean:
So, the geometric mean between and in this GP is .
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.