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In mathematics, the A.M, G.M, and H.M hold a special value as these three represent the average value of the particular series. All these means have a special purpose and definition in Math. Along with these three meanings, the relation between A.M and G.M also holds a special intention to outburst the inequality among these. This means that the inequality will further represent the list of non-negative real numbers that are commonly found in the A.M and are greater than or equal to the G.M present in the same list. However, the relation between A.M and G.M will only occur if the numbers of the list are the same in both situations.

What is Sequence in Math? 

When we talk about the relation between A.M and G.M, including the H.M, we conclude that it is a basic proportion of the mathematical sequence that helps to provide the basic knowledge of progressions. Further, the collection of objects in a special and unique field in Math is defined as the Mathematical sequence. 

In addition, the sequence is further defined as the process of progression from one state to another. The most common types of sequences available in Math include the Arithmetic Mean, Geometric Mean, and Harmonic Mean. 

However, the Arithmetic Mean describes a sequence of patterns of numbers in a unique way that helps to distinguish the consecutive terms between them while maintaining the constant nature throughout the overall sequence. On the other hand, the Geometric Mean represents the common ratio that is formed by the sequence of numbers with two consecutive terms of the same sequence. Lastly, the Harmonic mean represents the sequence’s progression where an Arithmetic Mean is formed with the reciprocal of terms occurring in a special order.

Definitions of A.M, G.M, and H.M

The elaborated definition of A.M, G.M, and H.M are further listed below to provide you with a better understanding. Ensuring everything about this is also important before understanding the relation between A.M and G.M. 

  • AM: The full form of AM is Arithmetic Mean. This means that it is also known as the average of the set of numbers present in a number series and is calculated by adding all the terms available in the set and then dividing the overall sum with the total numbers of terms present.

Formula: Arithmetic Mean = (x1 + x2 + …. + xn) / n or ∑ xi / n

                               here, xi = ith variable 

                                        n= number of variable on data set

  • GM: The full form of G.M is the Geometric Mean. This further represents the central term or the mean value present in the whole set of numbers. This is performed with the help of geometric progression. The Geometric Mean also denotes the ‘n’ terms present in the sequence and is calculated as the nth root of the whole series of terms in a sequence when obtained together. 

Formula: x̄geom =nni=1 xi  

                         =x1 .x2 .x3 ……………..xn

                      x̄geom = geometric mean

                      n= number of observation

                                   nni=1 xi    = nth square root of product of numbers.

  • HM: The full form of H.M is Harmonic mean and is obtained to get one of the types by calculating the average of the whole. Furthermore, it is calculated by dividing the number of values present in the series by the overall sum of the reciprocals of the terms present in the whole series. 

Formula: HM = n / [(1/x1) + (1/x2) + (1/x3) +…+ (1/xn)] 

x1, x2, x3,…, xn are the individual terms up to the nth term.

Explain the relation between A.M and G.M?

The relation between A.M and G.M along with the H.M is further classified with the help of the following:

Consider a, A.M, b is an Arithmetic Series. Now the common difference of the Arithmetic Progression is derived by:

AM-a= b- AM,

a+b= 2 AM…… (1)

Now, let’s consider a, G.M, b is a Geometric Mean, Then the common ratio of the GM will be: 

GM/a= b/GM

ab= GM2 ………. (2)

Now let’s assume in the Harmonic Mean, a, HM, b, here the reciprocals of every term will conclude the arithmetic progression, for instance:

1/a, 1/HM, 1/b  

Thus the common difference of the above AP is: 

1/HM – 1/a = 1/b- 1/HM, 

Similarly, 2/HM = 1/b + 1/a

2/HM = (a + b)/ab ………….(3)

Thus from the equation 1, 2, and 3, it can be assumed that, 

2/HM = 2AM/GM2

GM2 = AM x HM

Thus, in this way, the relation between A.M and G.M along with the H.M can be understood and well classified. 

Another example, for relation between A.M and G.M, is derived by considering two numbers a, and b whose values are greater than 0. Thus terms in the series represent a, and b, whereas the whole number of terms in the series represent n=2. Thus if AM, GM, and HM formula is used then the following can be derived:

AM = (a+b)/2,

GM = ab

HM = 21a+1b = 2a+bab =2aba+b 

Considering, (a+b)/2= AM and ab = GM2

HM= GM2 / AM

The relation between A.M and G.M,can thus also be written as, 

AM * HM = GM2,

or, GM = AM x HM 

Statistics and Properties of AM & GM

The statistics and Properties of AM & GM, along with the H.M are further listed below:

  • The Properties of AM & GM can be understood as this plays a vital role in playing major calculations. 
  • The Arithmetic Mean is very simple and easy to calculate and also has the tendency to provide the measures of the central group to obtain the grouped set of data. This is another property of the AM. 
  • The geometric mean further tends to calculate the stock indexes. Thus the property of GM is obtained as it can find out the portfolio’s annual returns. It is also used to understand the biological processes, including cell division, bacterial growth, etc.
  • Furthermore, the HM is used to understand and calculate the price of the earning in the form of ratio and other average multiples in terms of finance. Thus it can be considered that these are used to find out the Fibonacci sequence. 

Conclusion:

Thus, the relation between A.M and G.M can be obtained with the help of the formulae and can also play a vital role in the calculation of series. In mathematics, sequences often provide the equality condition of the inequality states with the help of the AM, GM, and HM. Also, the formulae can also be used to find out and calculate the Olympiad level inequality questions. Furthermore, in simple terms, the AM-GM denotes that for any kind of non-negative number, the arithmetic sequence will be higher than the geometric mean, or it can also become equal, but it will not become less than the geometric mean. Also, the proofs of the AM-GM are obtained with the help of a weighted average.