A.M, G.M and their Relationship

• We are already aware of the fact that the arithmetic mean (A.M) between two numbers a and b is ( are positives). If are arithmetic means between , then form an A.P.
the common difference since is the term of the A.P with as the first term.

and so on.
• The geometric mean (G.M) between two numbers is ( are positive)
If are the geometric means between , then are in G.P.

.
• If A and G be the A.M and G.M of two given positive numbers , then .

Example 1:

Show that the sum of the A.M's between two given numbers is times the single A.M between them.

Solution:

Let be the A.M's between . are in A.P.This A.P has terms.

Example 2:

If is the G.M between two quantities and are two A.M's between them, show that

Solution:

Let be the two quantities then, . If are two A.M's between , then, and

Example 3:

If A.M between two numbers is A and the G.M between them is G, find the numbers in terms of A & G.

Solution:

Let the numbers be

Then,

or

we know that

Example 4:

If m be the arithmetic mean of n consecutive integers, show that the sum of their cubes is

Solution:

Let the consecutive integers be

Their A.M =

Sum of the cubes of the consecutive integers.

Substituting for , we get

Example 5:

The sum of two numbers is 6 times their G.M, show that the numbers are in the ratio .

Solution:

Let be the two numbers.

Given

Squaring

We know that

Solving (1) & (2), we get,