The arithmetic mean and geometric sequence are important terms in the context of calculating financial and economic growth. Stock markets, increments, population growth, etc are the important areas that make use of these terms widely.

**Arithmetic Mean vs Geometric Sequence**

**The main difference between Arithmetic Mean and Geometric Sequence is that arithmetic mean is used to find the average out of the collection of numbers whereas geometric sequence is the mere collection of numbers with a constant ratio.**

The Arithmetic mean or simply the average is the collection of numbers divided by the count of numbers whereas the Geometric sequence is the collection of terms that is obtained by dividing or multiplying the constant term.

A sequence is the structured collection of terms in a repetitive pattern whereas ‘arithmetic mean’ is the average derived out of that sequence of numbers. The ‘arithmetic mean’ and ‘geometric sequence’ are the mathematical terms that are often used to find this methodical organization of terms.

The arithmetic mean is the average of the numbers in a sequence where the difference between two consecutive terms might or might not be separated by a constant number whereas when these terms are present in a definite ratio then the ratio is determined by the geometric sequence known as a common ratio.

## Comparison Table Between Arithmetic Mean and Geometric Sequence (in Tabular Form)

Parameters of Comparison | Arithmetic mean | Geometric sequence |
---|---|---|

Definition | The arithmetic mean is the average of the collection of numbers in a given sequence. | The geometric sequence is the collection of terms with the difference in the ratio of two consecutive terms being constant. |

Determined by | It can be determined by dividing the sum of the collection of numbers by the total count of numbers. | It can be determined by multiplying or dividing a constant to the preceding term. |

Form | This is expressed as an average of the collection. | This sequence is usually expressed in the exponential form. |

Common formula | A= (a1 + a2+ .. + an)/n (where a is the 1st digit and n is the total number of digits we can find the mean A through this formula) | tn = t1 . r(n – 1) (where r is the common ratio and tn is the nth term, t1 is the first term) |

Uses | The arithmetic mean or the average is used in the observational and experimental studies to get a brief idea of the large sample size because mean then becomes the central tendency of the data. | a geometric sequence is used in various sectors such as financial and economic sectors to calculate the growth rates, savings, costs, etc. |

## What is Arithmetic Mean?

The arithmetic mean is the average of the sequence of terms that may or may not be separated by the common difference. To find the mean we divide the sum of a collection of the terms with the total amount of numbers present. Average or the arithmetic mean is the simplest and convenient method to size down the large sample size because ‘mean’ is always the central tendency of any given data.

For the experimental research and the observational studies, mean can be calculated as the sum of the total number of observations divided by the number of observations which is written as:

Arithmetic Mean = (sum of all the observations)/(total number of observations)

When the data is present is a sequence then the average of any sequence can be determined by the given formula:

A= (a1 + a2+ .. + an)/n

‘A’ is the average or the arithmetic mean, ‘a’ is the 1^{st} term and ‘n’ is the total number of terms present in the collection

For example, we have to find the arithmetic mean of the sequence 2, 4, 6, 8, 10

This can be easily done by the aforementioned formula as: (2+4+6+8+10)/5= 6

The arithmetic mean has applications in daily life when observed. In the fields of anthropology, history, statistics, to calculate per capita income, etc, the average is of foremost importance. The arithmetic mean has certain limitations because it is just the approximate value and not the exact value. In the financial data where each figure of term matters, average can’t be used as a formula for the calculations.

## What is Geometric Sequence?

A geometric sequence is the sequence of numbers where consecutive terms are in common ratio. Simply when the progression is multiplied or divided by the same, non-zero number then the sequence obtained is called geometric.

This progression can be depicted as **a, ar, ar ^{2}, ar^{3}, ar^{ 4 }**and so on (where a is the 1

^{st}term and r is the common ratio)

For example: 3, 9, 27, 81, _ _ _

The geometric sequence is expressed in the exponential form by the formula: t_{n} = t_{1} ^{.} r^{(n – 1)} (where ^{ }t_{n} is the nth term, t_{1 }is the first term and d is the common ratio)

Geometrical sequences seem a bit more complex to figure out than the arithmetic mean but still have numerous uses in day to day works for example in calculating the growth rates, stock markets, interest rates, etc.

**Main Differences Between Arithmetic Mean and Geometric Sequence**

- The arithmetic mean is the average of the collection of terms by dividing the sum of the collection with the number of given terms whereas geographic sequence is the sequence of consecutive terms with the common ratio.
- The arithmetic mean can be obtained by adding the collection of terms and dividing them by the amount to terms present whereas the geometric sequence is obtained by multiplying or dividing the constant non-zero term from the preceding number.
- The arithmetic mean is the central tendency of the data set whereas the geometric sequence is exponential in variation.
- The arithmetic mean is largely used in observational studies, experiments, etc whereas geometric sequence is used in the stock markets, to calculate savings, cost, etc.
- The arithmetic mean is also used to size down the large data, to get the approximate hint of the results whereas geometric sequence is the sequence of exact data. The ‘arithmetic mean’, therefore can’t always provide the exact results.

## Conclusion

The arithmetic mean is the average of a collection of numbers where the common difference between the successive terms may or may not be defined by a constant whereas Geometric Sequence is simply the sequence of terms where the successive terms necessarily have common ratio defined by ‘r’.

The arithmetic mean is derived by dividing the sum of the collection of terms by the total number of terms in the series whereas the geometric sequence is derived by multiplying or dividing the successive terms with the common ratio.

The arithmetic mean is usually the central limit of any data whereas the geometric sequence is the exponential rise in the given collection of terms.

Both the arithmetic mean and geometric sequences have their applications in daily lives if we observe things around us. The arithmetic mean is used in various fields such as anthropology, experimental studies to determine the average value whereas geometric sequences are used in calculating the population growth, stock markets, etc

**References**

- https://www.tandfonline.com/doi/abs/10.1080/00029890.2001.11919815
- https://www.fq.math.ca/Scanned/22-4/schoen.pdf

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