Arithmetic Mean and Geometric Mean

The relationship between the arithmetic mean and geometric mean gives the arithmetic mean greater than the geometric mean of any two given numbers.

The arithmetic mean of two numbers is the mean value of the two numbers that satisfy the arithmetic progression series. The geometric mean of two numbers is the square root of the products of the numbers. In this article, we derive the relationship between the arithmetic mean and geometric mean and also understand the concept with solved problems.

Arithmetic Mean (AM)

Consider two numbers a and b; let a third number A be placed between a and b so that the sequence of a, A, and b is in an arithmetic progression series.

Here, number A is called the arithmetic mean of a and b

A – a = b – A,

Or, A = (a + b)/2

Example: Insert four numbers between 2 and 22 such that the resulting sequence makes an AP series.

Solution: Let the four numbers between 2 and 22 be

A1, A2, A3, and A4,

So far, we have a sequence as,

2,  A1, A2, A3, A4, 22.

From the above sequence,

a = 2, 

b = 22, and

n = 6,

The 6th term of the AP is given by,

22 = 2 + (6 – 1) .d 

d = 4.

Now, we can easily find out the values of the four terms,

A1 = a + d

= 2 + 4 = 6.

and, A2 = A1 + d,

= 6 + 4 = 10.

and, A3 = A2 + d,

= 10 + 4 = 14.

and, A4 = A3 + d,

= 14 + 4 = 18.

Therefore, the desired series is,

2, 6, 10, 14, 18, 22.

Geometric Mean (GM)

The geometric mean of two numbers, a and b, is given as the square root of the product of numbers.

The geometric mean can be expressed as,

GM = √(a.b).

For example, the geometric mean of 9 and 4 will be,

GM = √(9×4),

GM = √36

GM = 6.

Let us understand geometric mean with the help of an example,

Example: Add two such numbers between 1 and 729 that the resulting sequence forms a GP series.

Solution: Consider, G1 and G2 as two numbers between 1 and 729,

1, G1, G2, 729.

We know, r3 = 729,

r = 9,

We can now find G1 and G2,

G1 = ar,

G1 = 9.

And, G2 = ar2

G2 = 81.

Therefore, the desired sequence is,

1, 9, 81, 729.

Relationship Between Arithmetic Mean and Geometric Mean

Let us consider two real numbers, a and b.

Also, considering A and G are the AM and GM of a and b, respectively,

Then, A = (a + b)/2, and

G = √(a.b)

For, A – G,

(a + b)/2 – √(a.b),

Or, [(a + b) – 2 √(a.b)]/2

From, the equation (x – y)2 = x2 + y2 – 2xy, we get,

[ (√a – √b)2] / 2 ≥ 0

Or, A ≥ G.

Solved Examples

Relationship between Arithmetic Mean and Geometric Mean: Questions

Example 1: Consider two numbers,  a and b. Assume that the arithmetic means and geometric mean is 6 and √11, respectively. Find the value of a and b.

Solution: We have been given the values of AM and GM,

AM = (a + b) / 2 = 6. …(i)

GM = √(a.b) = √11. …(ii)

From equations (i) and (ii),

(a + b) = 12 …(iii)

a.b = 11 …(iv)

Implying the identity equation,

(a – b)2 = (a + b)2 – 4a.b,

(a – b)2 = 144 – 44,

(a – b)2 = 100,

And, a – b = 10. …(v)

Solving equations (v) and (iii), we get,

a = 11,

Adding this value of a in the equation we get,

b = 1

Therefore, the value of the desired numbers is 11, 1, or 1, 11.

Example 2: x and y are two real numbers. The arithmetic mean and the geometric mean of x and y are 5 and 3, respectively. Find the values of x and y.

Solution: We have been given the values of AM and GM,

AM = (a + b) / 2 = 5. …(i)

GM = √(a.b) = 3. …(ii)

From equations (i) and (ii),

(a + b) = 10 …(iii)

a.b = 9 …(iv)

Implying the identity equation,

(a – b)2 = (a + b)2 – 4a.b,

(a – b)2 = 100 – 36,

(a – b)2 = 64,

And, a – b = 8. …(v)

Solving equations (v) and (iii), we get,

a = 9,

Adding this value of a in the equation we get,

b = 1

Therefore, the value of the desired numbers is 9, 1, or 1, 9.

Conclusion

This article tells about the relationship between the arithmetic mean and geometric mean. For two given numbers, the relationship between the arithmetic mean and the following expression gives the geometric mean,

AM ≥ GM

The relationship establishes that for two numbers in a series, the arithmetic mean is greater than the geometric mean. The relationship between the arithmetic mean and geometric mean is established by subtracting the former with the latter. The identity equation for the relationship is given by,

(a – b)2 = (a + b)2 – 4a.b