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Sequences and Series, Geometric Mean (G. M.) , Relationship between a. M. And G. M
Geometric Mean (G. M.)
- If , , are in G. P. , then is called the geometric mean between and .
- If three numbers are in G. P. , the middle one is called the geometric mean between the other two.
- If are in G. P. ,
- Then are called n G. M. s between and .
- The geometric mean of numbers is defined as the root of their product.
- Thus if are numbers, then their
G. M.
Let be the G. M. between and , then are in G. P
Or or
- Given any two positive numbers and , any number of geometric means can be inserted between them Let be geometric means between and .
- Then is a G. P.
- Thus, being the term, we have
or or
Hence,
,
- Further we can show that the product of these G. M. s is equal to power of the single geometric mean between and .
- Multiplying , we have
Example:
Find the G. M. between and
Solution:
We know that if is the G. M. between and , then
G. M. between and
Relationship between a. M. And G. M
- Let and be the two numbers.
- Let and be the A. M. and G. M. respectively between and
Example:
The arithmetic mean between two numbers is and their geometric mean is . Find the numbers.
Solution:
Let the numbers be and . Since A. M. between and is ,
Since G. M. between and is ,
we know that
Adding and , we get,
Subtracting from , we get
or,
Required numbers are and .