Sequences and Series, Geometric Mean (G. M.), Relationship Between a. M. And G. M

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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 .

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