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Relation between AM and GM

Arithmetic Mean

The A.M or the Arithmetic Mean is referred to as the average of 2 given numbers. And if one A.M “A” is included between the two numbers which are “a” & “b”, then A is called the arithmetic mean present in between the numbers, a & b.

A.M. = A =

Here, the number A is called the Arithmetic mean of a & b numbers. And as per the property of A.M., you can state that A – a = b – A i.e. it’s called the value‘d’, which is the common difference of the AP given.

https://lh5.googleusercontent.com/PDD3RsF7pXYHxBeXXH6LSSZ3dzgKEK6MRxsrAPEPnWUL-9O6F5M-KGs7Q_qrSL7_DaZME5EJHTGl5wRoyp3TaWWzxvYP3EeMuWEVjEwtP6LflCzmjoN7oxZaeESRTzSMf8ShGFw

A.M is usually calculated for finding out sequence’s missing number between the 2 numbers given.

 Arithmetic Mean

The AM acts as the point of balance for the 2 given numbers.

Example

What is A.M of 8 and 64?

Given

a = 8

b = 64

Therefore, the A.M is equal to

AM = = = 36

As a result, the Arithmetic Mean is 10.

Now, is it possible to include 2 or more numbers in between the 2 numbers given to form the AP?

Assume A₁ , A₂ , A₃,…, An be n numbers between a and b in a way that a, A₁ , A₂ , A₃,…, An, b is the A.P.

Here, the number “b” = (n + 2)th term, that is, b = a + [(n + 2) – 1] d = a + (n + 1) d.

This results in the expression:

https://lh5.googleusercontent.com/WuEcNgx-GmrVdkV25Cf7GLjimqsswO_A6ojA9o_-sE3JsLK3O44Ijx015_lTWtHdU_KYeG5LCGmOK749P_R8dOwSWlXvoIrV5CueIQ9D2HfmsHJB4Lck6Pye2O08SSz_WgJ3ujk

Therefore, “n” numbers in between a & b are the following ones:

https://lh4.googleusercontent.com/Y8091RRw3BuBYa7tyxWMPUz7COXSWaZFi81HkN6kAzFVwpT1rbl_y0QepoDzbe0ijZ63xjfy9BtdDOSf1Sb9Y3OrsuNwQmPmYbJ-4e9IPcFtx85KAQn9rUH3tKcuJPDokquvWZ8

Example

To insert four numbers in between 4 and 20 such that the resulting sequence is an A.P.

Solution:

Let A₁, A₂, A₃, and A be the four numbers between 3 and 18 such that

4, A₁, A₂, A₃, 20 is the given sequence in the A.P.

Here, a = 4, b = 20, n = 5.

Hence,

20 = 4 + (5 –1) d

16 = 4d

d = 4

Therefore, A₁ = a + d = 4 + 4 = 8;

A₂ = a + 2d = 4 + 2 × 4 = 12;

A₃ = a + 3d = 4 + 3 × 4 = 16;

Hence, three numbers in between 4 and 20 are 8, 12, and 16

 

Geometric Mean

Geometric Mean or GM refers to a special average type of the 2 specified numbers. If you are having 2 numbers, then we’ll be multiplying the given numbers and later we’ll be taking the square root (√) of it.

If the two numbers are a & b, then the GM will be given be in the form:

X = √a.b

Geometric Mean

Example on GM:

Question: What’s the GM of 4 and 64?

Answer:

Given:

a = 4 and b = 64

GM =

=

= 16

Here, you can see that the sequence consists of 4,16, 64 which is the GP.

Now,

The Relationship between A.M. and G.M.

As seen above the A.M. & the G.M. formula, which are:

https://files.askiitians.com/cdn1/images/2017814-175457735-5797-equation-8.png

Where,

The 2 +ve numbers are a & b

Relationship between A.M. and G.M.

 

Assuming, A & G as A.M. & G.M. respectively

Therefore,

https://files.askiitians.com/cdn1/images/2017814-175511703-8205-equation-9.png

Let’s now do the subtraction of 2 of them

https://files.askiitians.com/cdn1/images/2017814-175526937-1933-equation-10.png

This represents that A ≥ G

Taking the Example

Question: Find the 2 numbers, If Arithmetic mean is 14 and the Geometric mean is 6 of the 2 positive real numbers.

Solution:

Given:

A.M. = = 14,

A.M = 28; ………..Eq 1.

G.M. = √a.b

6 = √a.b

a.b = 36………….Eq 2.

 

Now, putting these values in the numbers a & b in:

(a – b)2 = (a + b)2 – 4ab

(a – b)2 = (28)2 – 4(36)

(a – b)2 = 784 – 144

= 640

a – b = ± 80 ( by taking square root) …(3)

By solving (1st) and (3rd), we’ll be getting

a + b = 28

a – b = 80

a = 54, b = -26

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