Introduction
The world in which we live is made up of a variety of elements, including trees, clouds, rivers, mountains, buildings, homes, automobiles, different sorts of food, and religious beliefs. Nevertheless, people sometimes overlook the fact that numbers are the most crucial component in the maintenance of the system on our planet. In our lives, numbers are everywhere, whether it's a home number or a phone number. Numbers define us, from the number of properties we own to the number of marks we get in examinations to the amount of money we owe to the number of failures and achievements we have experienced.
Sequences are a collection of numbers that are ordered or defined by a given rule, indicating that they have something in common.
In mathematics, progressions are a collection of integers that are specified by a certain rule. It follows a certain formula to determine its terms.
In this case, the difference between the two is that while the sequence is based on a logical rule or statement, such as a set of prime numbers or a set of numbers divided by 2, the progression has a specific rule that must be followed for all of the elements of the set to be calculated. For example, 2, 4, 6, 8, 10, and 12 is a progression because the ratio between two consecutive numbers is 2 and we can calculate its nth term by using the 2n formula.
That encapsulates the reason why everyone should be familiar with and comprehend mathematics. Math is divided into many disciplines, with the two most important components being an arithmetic progression and arithmetic sequence.
Arithmetic Progression vs Arithmetic Sequence
The main difference between Arithmetic Progression and Arithmetic Sequence is that Arithmetic Progression is a series that has a common difference that is up to an nth term, while Arithmetic Sequence is a series that does not have a common difference up to an nth term. When the components of Arithmetic Progression are added together, they form an Arithmetic Sequence or an Arithmetic Series.
Arithmetic progression can be defined as any number of sequences within any range that result in a common difference in the result. Consider the following example: given a range of numbers ranging from 1, 2, 3, 4, — to any number, the difference between the number and its succeeding number would be common for any two numbers in this range: 1. AP is a series of integers in ascending order in which the difference between any two subsequent numbers is a constant value, also known as Arithmetic Progression (AP). It is referred to as Arithmetic Sequence in certain circles. A natural number sequence such as 1, 2, 3, 4, 5, 6,... is an Arithmetic Progression because the common difference between two succeeding terms (for example, 1 and 2) is equal to one (2 -1). This is true even in the case of odd and even numbers, as we can see that the common difference between two consecutive terms will be equal to 2.
An arithmetic Sequence is a collection of numbers or a range of numbers that follow a specified sequence of operations. It is possible to get a difference between two numbers in this sequence by subtracting one from the preceding number in the series. This difference would be universal to any two values in the range. Suppose you have a series of numbers from 5, 10, 15, and 20—now this sequence will have a common difference of 5. Sequences are often characterized by patterns that enable us to forecast what the following word will be.
You can see this in the sequence 3, 5, 7,... where you constantly add two to obtain the following term: 3, 5, 7,...
Although we only see a few phrases, the three dots after the sequence imply that the sequence may be expanded.
We may do this via the use of a pattern.
Example: The fourth term of the series should be nine, the fifth term should be eleven, and so forth.
Differences Between Arithmetic Progression and Arithmetic Sequence in Tabular Form
Parameter of Comparison | Arithmetic Progression | Arithmetic Sequence |
Concept | Arithmetic progression is a set of integers in a range that all have a common difference indicated by d, which is referred to as a common difference. This series continues until it reaches the nth word. | It is also known as Arithmetic series, or simply Arithmetic sequence, and it is the sum of components of Arithmetic progression that have a common difference given by d. |
Formula |
The following is the formula for Arithmetic Progression:
Let Ln be the nth term in the series of Arithmetic Progression; the nth term is determined in the following way: L1 + In = L2 + Ln-1 =... = Lk + Ln-k+1 = Lk + Ln-k+1 Ln = 12(Ln-1 + Ln+1) Ln = L1 + (n – 1)d, where n is one, two, three, four, or more |
The following is the formula for Arithmetic Sequence or Arithmetic Series: Let M be the sum of all the numbers.
M = 12(L1 + Ln)n M = 12(2L1 + d(n-1)n M = 12(2L1 + d(n-1)n M = 12(2L1 + d(n-1)n |
Uses |
Calculating balance sheets and doing monetary operations are all examples of when Arithmetic Progression is employed. It is also used in banking. Finance-related services are examples of when this term is used.
In addition, it is employed in architecture and construction. | It is used in architecture, building, construction of equipment, and other items with precise dimensions. It is also used in finance and banking to calculate interest rates and other financial information. |
Range |
Arithmetic Progression is a sequence of terms that may be in any range up to the nth term that is defined.
This series has a common difference, which may be calculated by subtracting a number from the number immediately before it. | The term "Arithmetic Sequence" or "Arithmetic Sequence" refers to a series of numbers with a range from one to infinity. |
Differences | Arithmetic Progression is a technique for determining a missing term or the nth term in a series by determining the common difference between the terms in the series as a whole. | To figure out the total, Arithmetic Sequence or Arithmetic Series is used. It takes the components of Arithmetic progression such as the nth term and the common difference and adds them all together. |
What is Arithmetic Progression?
Arithmetic progression is defined as a series of integers in which, for every pair of successive terms, we get the second number by multiplying the first value by a constant, and vice versa. The common difference (C.F.) of the arithmetic progression is the constant that must be added to any term of an AP to get the following term.
Calculated progression (AP) is an integer collection in which the difference between any two successive values in the series is always equal to one. Arithmetic progression (AP) is also known as arithmetic sequence (AS). Examples are the numbers 3, 6, 9, 12, 15,..., and 30. Each subsequent number is three digits higher or lower than the one that came before it. To test whether or not the number series is in APR, you must first ascertain whether or not the difference between all of the terms is constant before proceeding.
Our everyday lives are filled with instances of Arithmetic progression if we pay careful attention to what is going on around us. For instance, the number of pupils in a class, the number of days in a week, or the number of months in a year are all examples of quantitative measures. Progressions are a mathematical term that refers to this pattern of series and sequences as a whole.
AP is a series of integers in ascending order in which the difference between any two subsequent numbers is a constant value, also known as Arithmetic Progression (AP). It is referred to as Arithmetic Sequence in certain circles. A natural number sequence such as 1, 2, 3, 4, 5, 6,... is an Arithmetic Progression because the common difference between two succeeding terms (for example, 1 and 2) is equal to one (2 -1). This is true even in the situation of odd and even numbers since we can see that the common difference between two consecutive terms will be equal to 2.
If we look at ourselves in our daily life, we will see that we come across Arithmetic progression fairly often. For example, the number of pupils in a class, the number of days in a week, or the number of months in a year. In mathematics, this pattern of series and sequences has been extended as a progression pattern.
What is Arithmetic Sequence?
When it comes to numbers, an arithmetic sequence is a list of integers where the difference between subsequent terms is always the same. An arithmetic series may begin with any number, but the difference between succeeding terms must always be the same, regardless of where the sequence begins.
Arithmetic Sequence or Arithmetic Series is the sum of components of Arithmetic Progression that have a common difference and an nth term. It is also known as Arithmetic Series. To calculate the total, first, the first and final terms of the series are put together, then the sum of these terms is multiplied by 12 and the resulting is multiplied by the number of terms in the series
Take, for example, a sequence of numbers such as 4, 8, 12, 16, and nth. The first term is indicated by L1, and the nth term is denoted by Ln. When you add L1 and Ln together, the total of these terms will be multiplied by 12 and the number of terms in the series will be determined.
The phrase Arithmetic Series refers to a sequence of integers in which the difference between any two successive terms is always the same, regardless of how many terms are in the sequence. To put it another way, it signifies that the next number in the series is computed by multiplying the previous number by a predetermined integer. In addition, an Arithmetic Sequence may be expressed as follows:
a, a + d, a + 2d, a + 3d, a + 4d are all possible combinations.
where a represents the first word
d is the most frequent difference between two words in a sentence.
Consider the following sequence: 5, 11, 17, 23, 29, 35,... The constant difference is 6 in each of the numbers from 5 to 11.
Main Differences Between Arithmetic Progression and Arithmetic Sequence in Points
- In mathematics, an arithmetic progression is a sequence of values within a specific range that has a common difference that is consistent and can be obtained by subtracting two components from the series.
- Arithmetic Sequence is the sum of the components in a series of Arithmetic Progression that is produced by adding them together.
- Accounting, financial, and monetary circumstances, as well as certain building situations, are all examples of when Arithmetic Progression is used.
- When it comes to construction and building, and particularly in architecture, the Arithmetic Sequence is applied.
- It is possible to utilize Arithmetic Progression to find out the nth term and common difference, but Arithmetic Series may be used to figure out the sum of the components of Arithmetic Progression.
- Arithmetic Progression is a sequence of terms that may be in any range up to the nth term that is defined. In contrast, an Arithmetic Sequence or Arithmetic Series consists of a series of numbers with a common difference that can be calculated by subtracting a number from its preceding number, whereas this series has a common difference that can be calculated by subtracting a number from its preceding number.
- Let Ln be the nth term in the series of Arithmetic Progression; the nth term is determined in the following way: L1 + Ln = L2 + Ln-1 =... = Lk + Ln-k+1 = Lk + Ln-k+1 Ln = 12(Ln-1 + Ln+1) where Ln = 12(Ln-1 + Ln+1) Ln = L1 + (n – 1)d, where n is 1, 2,..., but the formula for Arithmetic Sequence or Arithmetic Series is: Ln = L1 + (n – 1)d, where n is 1, 2,..., Let M indicate the total of the parts. M is equal to 12(L1 + Ln)n. M is equal to 12(2L1 + d(n-1))n.
- Using the common difference from the series, Arithmetic Progression can be used to find a missing term or the next term in a series, whereas Arithmetic Sequence or Arithmetic Series can be used to find the sum by taking the elements of Arithmetic progression, such as the nth term and the common difference, and combining them.
Conclusion
It doesn't matter whether it's Arithmetic Progression or Arithmetic Sequence; both are important parts of mathematics that help us in our daily lives in a variety of ways, whether it's financial calculating or situations with details, such as in an architect's office or during the construction of a building or object that requires precise lengths and diameters, both are important parts of mathematics. The study of arithmetic may be beneficial in a variety of ways since the world would be meaningless without the numbers that we utilize in our daily lives.
References
- https://www.cs.umd.edu/~gasarch/TOPICS/vdw/heathbrown.pdf
- https://people.math.gatech.edu/~trotter/papers/56.pdf