Relation between A.M., G.M, and H.M

Introduction

The relation between AM , GM and HM is written as $\mathrm{AM\times\:HM\:=\:GM^{2}}$ . When studying sequences in math, we also encounter the relationship between AM, GM, and HM. These three represent the mean or average of the corresponding series. The Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are all abbreviations for mean. The mean of the arithmetic progression, the geometric progression, and the harmonic progression is represented by AM, GM, and HM, respectively. One should be familiar with these three meanings and their formulas before learning about how they relate to one another.

What is Arithmetic Progression?

An arithmetic progression (AP) can be defined in one of two ways −

• A series in which the differences between each pair of the following terms are the same is known as an arithmetic progression.

• An arithmetic progression is a sequence in which every term—aside from the initial term—is created by adding the previous term by a predetermined amount.

As an illustration, 2, 5, 8, 11, 14, 17,20, 23, and 26 have

• a = 1 (the first term)

• d = 3 (the "common difference" between terms)

An arithmetic sequence is typically written as follows: { a, a+d, a+2d, a+3d,... } etc. The following equation is obtained using the example given above − $\mathrm{\lbrace\:a\:,\:a\:+\:d\:,\:a\:+\:2d\:,\:a\:+\:3d\:.....\rbrace\:=\:\lbrace\:2\:,\:2\:+\:3\:,\:2\:+\:2\times\:3\:,\:2\:+\:3\times\:3\:......\rbrace\:=\:\lbrace\:2\:,\:5\:8\:,\:11\:......\rbrace}$

What is Arithmetic Mean?

The mean or arithmetic average is another name for the arithmetic mean. It is determined by adding up each number in a particular data set, then dividing the result by the overall number of items in the data set. For uniformly distributed integers, the middle number serves as the arithmetic mean (AM).

If the arithmetic progression or a group of values is $\mathrm{a_{1}\:,\:a_{2}\:,\:a_{3}\:......\:a_{n}}$

then

$$\mathrm{AM\:=\:\frac{(a_{1}\:+*\:a_{2}\:+\:a_{3}\:........\:+\:a_{n})}{n}}$$

So let's take an example, so the given AM series is {4, 6, 8, 10, 12}. So to find the arithmetic mean,, we would find the sum of the AM series ,i.e., {4 + 6 + 8 + 10 + 12 = 40} and divide it by the number of values in the series ,i.e., { n = 5}. Hence, the arithmetic mean is 40/5 = 8

What is Geometric Progression?

A unique kind of progression called a geometric progression is one in which the succeeding phrases share a common ratio that is always the same. It also goes by the name GP. The general representation of the GP is in the form $\mathrm{a\:,ar\:,\:ar^{2}}$ where r is the common ratio of the progression and ‘a’ is the first term. It is possible for the common ratio to have both negative and positive values.

Example − A GP with a common ratio of 3 is 1, 3, 9, 27, 81, etc.

To find the $\mathrm{n^{th}}$ term of the GP, we can use the following formula −

$$\mathrm{a_{n}\:=\:ar^{n\:-\:1}}$$

Where

• a is the first term of the GP

• r is the common ratio

• n is the number of values in the GP series.

What is Geometric Mean?

The Geometric Mean (GM) is the average value or mean that, by taking the product of a set of numbers' values as its root, indicates the set's central tendency. In essence, where n is the total number of values, we multiply all 'n' values together and subtract the $\mathrm{n^{th}}$ root of the numbers.

If the geometric progression or a group of values is $\mathrm{a^{1}\:,\:a^{2}\:,\:a^{3}\:,\:......\:a^{n}}$

then

$$\mathrm{GM\:=\:\sqrt[n]{(a^{1}\times\:a^{2}\times\:a^{3}\times\:......\times\:a^{n})}}$$

or

$$\mathrm{GM\:=\:(a^{1}\times\:a^{2}\times\:a^{3}\times\:......\times\:a^{n})^{1\:/\:n}}$$

For example, the geometric mean for the GP series { 6, 12}, is equivalent to $\mathrm{\sqrt{(6\times\:12)}\:=\:\sqrt{72}\:=\:6\sqrt{2}}$

What is Harmonic Progression?

The reciprocal of the terms in the arithmetic progression is used to create the harmonic progression. The terms of the harmonic progression are 1/a, 1/(a + d), 1/(a + 2d), 1/(a + 3d), 1/(a + 4d), etc., if the provided terms of the arithmetic progression are a, a + d, a + 2d, a + 3d,.... Here, the first word is a, and the common difference is d. The values of a and d are both non-zero.

To find the $\mathrm{n^{th}}$ term of the HP, we can use the following formula −

$$\mathrm{a_{n}\:=\:\frac{1}{(a\:+\:(n\:-\:1)d)}}$$

Where,

• “a” is the first term of A.P

• “d” is the common difference

• “n” is the number of terms in A.P

What is Harmonic Mean?

The harmonic mean is a measure of central tendency. Consider the situation where we wish to identify a single value that may be utilized to characterise the behavior of data around a central value. Then, such a value is referred to as a central tendency measure.

If a collection of observations is given by $\mathrm{h_{1}\:,\:h_{2}\:,\:h_{3}\:......\:h_{n}}$. This data set's reciprocal terms are $\mathrm{1/h_{1}\:,\:1/h_{2}\:,\:1/h_{3}\:......1/h_{n}}$. As a result, the harmonic mean formula is as follows −

$$\mathrm{HM\:=\:\frac{n}{(\frac{1}{h_{1}}\:+\:\frac{1}{h_{2}}\:+\:.......\:+\:\frac{1}{h_{n}})}}$$

For example, the harmonic mean for the HP series { 1/6, 1/8, 1/10}, is equivalent to 3/(1/6 + 1/8 + 1/10) = 3/(1/6 + 1/8 + 1/10) = 40

What is the relationship between the three means?

The argument that the result of AM is greater than the result of GM and HM can be used to understand the relationship between AM, GM, and HM. The following expression can be used to express the relationship between AM, GM, and HM.

$$\mathrm{AM\:>\:GM\:>\:HM}$$

The formula of relation in between AM, GM and HM is

$$\mathrm{AM\:>\:GM\:=\:GM^{2}}$$

Now let's see how this formula is derived for better understanding.

Let’s say we have an arithmetic progression a, AM, b

Here, the common difference is

$$\mathrm{a\:-\:AM\:=\:AM\:-\:b}$$

$$\mathrm{a\:+\:b\:=\:2AM\:..........(i)}$$

Now let's say we have a GP(Geometric Progression)

with common ratio $\mathrm{GM/a\:=\:b/GM}$

$$\mathrm{ab\:=\:GM^{2}\:.......(ii)}$$

Next, the harmonic progression a, HM, n and the reciprocal of these terms will result in an arithmetic progression,i.e.,

$$\mathrm{1/a\:,\:1/HM\:,\:1/b}$$

with a common difference

$$\mathrm{1/a\:-\:1/HM\:=\:1/HM\:-\:1/b}$$

$$\mathrm{2/HM\:=\:1/b\:+\:1/a}$$

$$\mathrm{2/HM\:=\:(a\:+\:b)/ab\:........(iii)}$$

Now put equation (i) and equation (ii) into equation (iii).

$$\mathrm{2/HM\:=\:2AM/GM^{2}}$$

$$\mathrm{AM\times\:HM\:=\:GM^{2}}$$

So that's how we get the relationship between AM, GM, and HM.

Solved Examples

1) Determine the geometric mean (GM) value if the harmonic mean (HM) is 56/9 and the arithmetic mean (AM) is 9.

Answer −

Given AM (Arithmetic Mean) − 9

Given HM (Harmonic Mean) − 56/9

And we have to find the geometric mean.

So to do that, we can use the relationship formula between AM, GM, and HM

$$\mathrm{AM\times\:HM\:=\:GM^{2}}$$

$$\mathrm{9\times\:56/9\:=\:GM^{2}}$$

$$\mathrm{GM^{2}\:=\:56}$$

$$\mathrm{GM\:=\:\sqrt{56}}$$

$$\mathrm{GM\:=\:2\sqrt{14}}$$

Thus, 2√14 is the required geometric mean.

Conclusion

• The arithmetic mean is determined by adding up each number in a particular data set, then dividing the result by the overall number of items in the data set.

$$\mathrm{AM\:=\frac{(a_{1}\:+\:a_{2}\:+\:a_{3}\:.......\:+\:a_{n})}{n}}$$

• A unique kind of progression called a geometric progression is one in which the succeeding phrases share a common ratio that is always the same.

• The Geometric Mean (GM) is the average value or mean that, by taking the product of a set of numbers' values as its root, indicates the set's central tendency. $\mathrm{GM\:=\:\sqrt[n]{(a^{1}\times\:a^{2}\times\:a^{3}\times\:......\times\:a^{n})}}$

• The reciprocal of the terms in the arithmetic progression is used to create the harmonic progression

• The harmonic mean is a measure of central tendency. $\mathrm{HM\:=\:\frac{n}{(\frac{1}{h_{1}}\:+\:\frac{1}{h_{2}}\:+\:.....\:+\:\frac{1}{h_{n}})}}$

• The formula for the relationship between AM, GM, and HM.

$$\mathrm{AM\times\:HM\:=\:GM^{2}}$$

FAQs

1. What relation Exists Between AM and GM/HM?

The equation $\mathrm{AM\times\:HM\:=\:GM^{2}}$ can be used to show the relationship between AM, GM, and HM. Here, the square of the geometric mean is equal to the product of the arithmetic mean (AM) and harmonic mean (HM) (GM).

2. In an arithmetic progression, how do you find d?

We will determine the difference between any two successive terms of the AP in order to determine the d in an arithmetic progression

3. What is the Geometric Progression Common Ratio?

Determining the ratio of any term to its preceding term yields the common ratio. Take the G.P, 1, 3, 9, for instance. R = 9/3 = 3 is the common ratio.

4. What Does Arithmetic Mean In Practice?

Central tendency is measured using the arithmetic mean. By taking into account all of the data, it enables us to determine the frequency distribution's centre.

5. What is the formula for harmonic progression?

Finding the $\mathrm{n^{th}}$ term in a harmonic progression is the definition of the harmonic progression formula. The harmonic progression $\mathrm{n^{th}}$ term is equal to $\mathrm{1/(a\:+\:(n\:-\:1)d)}$.