# 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,