Geometric mean and arithmetic progression are two different topics of mathematics that have immense real-life applications. Geometric mean is one of the measures of central tendency apart from mean, median, mode, and harmonic mean. It possesses some mathematical properties like arithmetic mean and harmonic mean. Geometric mean is based on all the observations and is rigidly defined.

They are difficult to compute and comprehend but are extremely useful in real life. Likewise, arithmetic progression is based on sequences and series and is beneficial in commercial problems and a variety of different things. In this article, we will try to understand one by one the concepts of arithmetic progression and geometric mean in detail.

## Geometric Mean

Geometric mean can be defined as the n-th root of the product of a set of n positive observations. The formula for geometric mean can be given as follows:

G = (x_{1} × x_{2} × x_{3} × …… × x_{n})^{1/n}

For a grouped frequency distribution, the geometric mean is given by

G = (x_{1} ^{f1} × x_{2} ^{f2} × ….. × x_{n} ^{fn})^{1/N}

Where N = Σ f.

## Properties of Geometric Mean

- If all the observations assumed by a variable are constants, say k, where k > 0, then the geometric mean of the observations is also k.
- Geometric mean of the product of two variables is the product of their geometric mean which means that if z = x × y, then Geometric mean of z = (Geometric mean of x) × (Geometric mean if y)
- Geometric mean of the ratio of two variables is the ratio of the geometric mean of the two variables which means that if z = x/y, then Geometric mean of z = (Geometric mean of x) / (Geometric mean of y).

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## Arithmetic Progression

Before talking about arithmetic progression, we need to understand what sequence is. An ordered set of numbers a_{1}, a_{2}, a_{3}, a_{4},.…… , a_{n }is a sequence if according to some definite law or rule, there is a definite value of an, which is called the term or element of the sequence, corresponding to any value of the natural number n. A sequence a_{1}, a_{2}, a_{3}, ……, a_{n} is called an arithmetic progression (A.P.) when a_{2} – a_{1} = a_{3} – a_{2} = a_{4 }– a_{3} = ….. = a_{n} – a_{n -1}. This means that arithmetic progression is a sequence in which each term is obtained by adding a term that is constant (d) to the preceding term. This constant (d) is known as the common difference of the arithmetic progression. Let us take a few examples that will help to understand it clearly.

- 2, 4, 6, 8, 10, … is an arithmetic progression in which d = 2 is a common difference.
- 10, 8, 6, 4, 2,…. is an arithmetic progression in which d = -2 is a common difference.

## Important Properties of Arithmetic Progression

- Generally, an arithmetic progression series can be written as a, a + d, a + 2d, a + 3d,……
- To find the nth term of an A.P., we use the following formula:

nth term (t_{n}) = a + (n – 1) d,

where n is the position number of the term you want to obtain and a is the first term of the arithmetic progression.

- To find the sum of the n terms of an arithmetic progression, we use the following formula:

s = n/2 [ 2a + (n – 1) d]

- If the first and the last term of an arithmetic progression is known to us, we can find the sum of total terms by using the following formula:

s = (n/2) a + l

where a is the first term and l is the last term of an arithmetic progression.