The world we live in is made up of many things whether it is the trees, clouds, rivers, mountains, buildings, houses, vehicles, types of food, and religion. But people often forget to mention the most important component that maintains the system in this the world and that component is numbers. Numbers are present everywhere whether it’s a house number, phone number, numbers define us from the number of properties we have to number of marks we get in exams to the amount of wealth we have to even the number of failures and successes.

That defines the reason why everybody needs to learn and understand math. Math has various branches and the two main components of math are arithmetic progression and arithmetic sequence.

**Arithmetic Progression vs Arithmetic Sequence**

The difference between Arithmetic Progression and Arithmetic Sequence is that Arithmetic Progression is a series that has a common difference which is up to an nth term. Arithmetic Sequence or Arithmetic Series is the sum of the elements of Arithmetic Progression.

Arithmetic Progression is any number of sequences within any range which gives a common difference. For example, take a range from 1,2,3,4, — to any number now the difference between the number and its succeeding number would be common for any two numbers in this range.

Arithmetic Sequence is a group of numbers or ranges of numbers with definite sequence. If a number in this sequence is subtracted by its previous number that we would get a difference which would be common with the difference of any two numbers in this range. For example, take a sequence from 5,10,15,20— now this sequence will have a common difference of 5.

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

Parameter of Comparison | Arithmetic Progression | Arithmetic Sequence |
---|---|---|

Concept | Arithmetic Progression is a series of numbers in a range that has a common difference denoted by d. This series extends to an nth term. | Arithmetic Sequence or Arithmetic series is the sum of elements of Arithmetic progression having a common difference denoted by d. |

Formula | Formula used for Arithmetic Progression is: Let Ln denote the nth term in the series of Arithmetic Progression, it is calculated as follows: · L1 + Ln = L2 + Ln-1 = … = Lk + Ln-k+1 · Ln = ½(Ln-1 + Ln+1) · Ln = L1 + (n – 1)d, where n is 1, 2, … | Formula used for Arithmetic Sequence or Arithmetic Series is: Let M denote the sum · M = ½(L1 + Ln)n · M = ½(2L1 + d(n-1))n |

Uses | Arithmetic Progression is used in Banking, Accounting, and to calculate balance sheet and used in monetary work. Used in services related to finance. Also used in architecture and building. | Arithmetic Sequence or Arithmetic Series is used in architecture, building, construction of machinery, and other things with accurate diameters also used in finance and banking. |

Range | Arithmetic Progression consists of a series of any range up to the nth term. This series has a common difference deduced by subtracting a number from its preceding number. | Arithmetic Sequence or Arithmetic Series consists of a series of a range up to infinity. |

Differences | Arithmetic Progression is used to find out a missing term or the nth term of that particular series by finding out the common difference from the series. | Arithmetic Sequence or Arithmetic Series is used to find out the sum by taking the elements of Arithmetic progression like the nth term, common difference. |

## What is Arithmetic Progression?

Arithmetic Progression is a sequence or range of elements that are used to calculate various terms like common difference and nth term. The common difference should be common for every element in the series which is subtracted by its previous element to be called an Arithmetic Progression Series.

For example, take a series like 3,6,9,12—-nth term, now when you subtract 3 from 6 or subtract 6 from 9 and so on you get the common difference 3, this tells us that the series is an Arithmetic Progression as the common difference is consecutive.

## What is Arithmetic Sequence?

Arithmetic Sequence or Arithmetic Series is the sum of elements of Arithmetic Progression having a common difference and an nth term. To calculate the sum first term and the last term of the series are added than the sum of these terms is multiplied by ½ and the resultant is multiplied by the number of terms in the series.

For example, take a series like 4,8,12,16—nth, now L1 is the first term and the nth term can be denoted by Ln. Add L1 and Ln and the sum of these terms will be multiplied by ½ and the number of terms in the series.

**Main Differences Between Arithmetic Progression and Arithmetic Sequence**

- Arithmetic Progression is the series of a specified range that has a common difference which is consistent getting by subtraction of two elements in the series.
- Arithmetic Sequence is the sum that is obtained by the elements of a series of Arithmetic Progression.
- Arithmetic Progression is used in Banking, finance, and monetary situations and also in some situations related to construction.
- Arithmetic Sequence is used in situations of construction and building and mainly architecture.
- Arithmetic Progression can be used to find out the nth term and common difference whereas Arithmetic Series is used to find out the sum of elements of Arithmetic Progression.

## Conclusion

Whether it is Arithmetic Progression or Arithmetic Sequence both are an important part of math which help us in our daily life in many ways whether it is financial calculating or situations with details whether in an architect or in construction of any building or object which requires detailed lengths and diameters arithmetic can help us in countless ways as the world is nothing without the numbers that we use in our everyday life.

## References

- https://www.cs.umd.edu/~gasarch/TOPICS/vdw/heathbrown.pdf
- https://people.math.gatech.edu/~trotter/papers/56.pdf
- https://augusta.pure.elsevier.com/en/publications/avoiding-arithmetic-progressions-mod-m-and-arithmetic-progression

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