Introduction
Math is not only a subject of study; rather, it is a philosophy to be lived by. A part of the routine that we do every day. The ideas and procedures of mathematics may be found in every facet of our daily lives. It is possible to make several broad conclusions from the results of an investigation based on the identification of recurring patterns of occurrence. Calculus applies to the overwhelming majority of real-world situations.
The answer to this conundrum lies in the linear function. There is a possibility that some of the real-time samples will include examples that are based on progressions and series. In an arithmetic progression, the next number is obtained by adding the current value to the value of the preceding number. To construct a connection between the several equations that make up the solution, a linear function is used.
Arithmetic Sequence vs Linear Function
Mathematical sequences and linear functions are distinct in that they are both polynomial functions; however, they differ in that an arithmetic sequence is an increasing or decreasing succession of integers with the same constant difference. Linear functions and mathematical sequences are distinct in that they are both polynomial functions.
The phrase "arithmetic sequence" refers to a series of real numbers in which each term is the sum of the one before it and a constant (called the common difference). For instance, if we begin with 1 and use a common difference of 4, we get the following finite arithmetic sequence: 1, 5, 9, 13, 17, 21. Additionally, we obtain the following infinite arithmetic sequence: 1, 5, 9, 13, 17, 21, 25, 29, etc., 4n+1,......................
An algebraic equation is said to be linear if each term in the equation is either a constant or the product of a constant and (the first power of) a single variable. Linear equations are the simplest kind of algebraic equations. Linear equations might include one, two, three, or even more than three variables.
Arithmetic sequences and linear functions both have a rate of change that remains the same across time. Their domains are not the same, which is one of the reasons why they are distinct from one another. Linear functions are defined for all real numbers, while arithmetic sequences are specified for natural numbers or a subset of natural numbers.
Difference Between Arithmetic Sequence and Linear Function in Tabular Form
| Parameters | Linear Algebra | Arithmetic sequence |
| Branch of Maths | Calculus and linear algebra both make heavy use of it. | It is used in routine, elementary-level mathematical operations. |
| Values | Consistent values may be found here. | It is not possible to acquire the constant values. |
| Plotting of Graph | It's impossible to get anything except a straight line. | Both the positive and negative sides of the graph may be shown here. |
| Application | To figure out how much room there is. | To figure out how many objects there are. |
| Area | When we use the plot to compute the area, we will get the same result every time. | The area varies from one to the next when it is computed. |
What is Arithmetic Sequence?
A series of numbers may be referred to as an arithmetic sequence or an arithmetic progression. The common denominator is a term that refers to the difference that exists between each integer in an arithmetic series. In an arithmetic series, there will never be a situation in which two consecutive numbers have the same value. For anything to be considered a sequence, it has to follow a certain pattern from beginning to end.
The difference that is always there between two integers is referred to as the "common difference," and it is denoted by the phrase. It is represented by the letter 'd' in the alphabet. One thing is different throughout the whole series. Utilizing the principle of common difference allows one to go from one number to another. Because we may get the numbers that came before in the series by either subtracting or adding the common difference with the previous number. This is the process through which everything comes together.
If the difference in meaning between the following two words is positive, then we say that the sequence is ascending. A sequence is said to have a declining sequence if there is a decreasing difference between each succeeding word.
A "finite arithmetic progression" is a kind of arithmetic progression that has a set number of stages rather than an infinite number of steps. There are a variety of distinct formats for arithmetic series to choose from. How does the mathematical progression act when the common difference is taken into account? Two distinct classifications may be applied to infinity. In an arithmetic progression, it is conceivable to have an infinite with a negative sign, but it is not feasible to have an infinity with a positive sign.
- The common difference must be a positive one to reach positive infinity.
- If the common difference is negative, then the members of the series will become closer and closer to negative infinity.
In our day-to-day lives, we often make use of mathematical sequences as a rule of thumb. Using a paper roll, as an illustration, the "starting terms" for this example include both the diameter of the roll and the thickness of the paper. Both of these dimensions are referred to as "role diameter." It is possible to find the whole role in this manner. There are a variety of other applications for it.
- When setting up a theater's seating chart, the mathematical progression method is often used.
- There is a logical process that may be used to determine the length of the rungs of a ladder.
- When a basketball game's point total increases by one, an additional point is likewise added to the player's AP.
A series in which each term rises by adding or subtracting some constant k is referred to as an arithmetic sequence. In contrast to this is a geometric sequence, in which each term grows by dividing or multiplying some constant k.
An arithmetic sequence is a series of integers that are arranged in a certain order and share a distinguishing characteristic with the preceding and subsequent terms. For example, in the mathematical sequences 3, 9, 15, 21, and 27, the difference that occurs most often is the number 6. One kind of arithmetic sequence is referred to as an arithmetic progression.
The common difference in the arithmetic sequence remains the same between any two words that are sequential to one another. First, let's have a look at the definition of a series. Sequences are the names given to groups of numbers that proceed in a predetermined order. An arithmetic sequence may be defined as a series in which each number in the series is obtained by adding 5 to the value of the term that came before it. There are two different formats for the formulas that make up the arithmetic sequence.
This formula, referred to as the nth term formula, may be used to get the nth term of an arithmetic series.
The formula is used to calculate the total number of terms in the first n steps of an arithmetic sequence.
Any term in the arithmetic sequence may be found by using the appropriate formula, which can be found here. Let's look at the definition and the formulas of arithmetic sequences, as well as their origins and numerous examples, so that we may have a better understanding of what they are.
There are two methods to define what is known as an arithmetic sequence. In an arithmetic sequence, "every term is generated by adding a fixed number (positive or negative or zero) to its preceding term." This is referred to as a "series where the differences between every two succeeding terms are the same." The steps that follow form an arithmetic sequence since each term is generated by adding a constant number four to the phrase that came before it.
What is Linear Function?
At the moment, the term "linear function" is used in both of these subfields of mathematics. To be more explicit, the subjects of Calculus and Linear Algebra, respectively, are being discussed here. Calculus always produces a straight line when plotting the graph of a linear function. This article describes a polynomial function that has a graph that is a straight line and a degree that is either one or zero. Analytical techniques such as functional analysis and mathematical analysis both make use of the linear function. In this particular instance, the route follows a straight line.
In calculus and analytical geometry, a polynomial of degree one or less is considered to be the Linear Function. This collection includes further polynomials with zero degrees of degree. When the degree of the polynomial is equal to zero, linear functions are said to be constant. When this constant function is presented on the graph, there is a line drawn across the horizontal axis.
It is possible, with the use of linear algebra, to determine the area of any given space by employing a linear function. A third term is produced as a direct consequence of the link that is made between the two locations. It's possible to discover this application when you're plotting the graph of speed, time, and distance.
A function on the coordinate plane that represents a straight line is referred to as a linear function. One example of a linear function is the equation y = 3x - 2, which represents a straight line when plotted on a coordinate plane. Since y may be replaced by f in this function, it is possible to write it as f(x) = 3x - 2, as seen above (x).
The term "linear function" is used in the field of mathematics to refer to two distinct ideas, both of which are related to one another.
In calculus and related subjects, the phrase "linear function" refers to any polynomial function whose graph is a straight line. This word is also used to refer to linear functions. When attempting to distinguish one kind of linear function from another, it is common to practice making use of the phrase "affine function."
In linear algebra, mathematical analysis, and functional analysis, a linear function is a linear map that represents a linear relationship.
A function is said to be linear if the function's graph can be represented by a straight line. This indicates that the function just makes use of one or two variables and does not make use of any mathematical exponents or powers. For the function to continue to be linear, any extra variables that it takes into account must either be constants or variables whose values are already known.
You might try making a list of characteristics that the function has to satisfy to determine whether or not it is a linear function.
The function must have at least one and no more than two real variables as its initial demand. A system may only ever have one unknown, which is referred to as the unknown variable. If the function C = 2 * pi * r is linear, then the only real variables are C and r, whereas pi remains a constant throughout the calculation.
By the second rule of thumb, variables may not have any exponents or powers linked to them. You can just cut them into squares; there is nothing more you can do with them. Every variable must be accounted for inside the denominator.
The very last criterion is that the function must produce a straight line when it is plotted. If the function exhibits any kind of curve, it cannot be considered legitimate.
As a consequence of this, the graph of any linear function will always seem like a straight line. It makes no difference whether the line is slanted or straight; it may go up or down, left or right, or any combination of those directions. It does not make a difference where on a graph a function is shown as long as the line that connects the function points in the same direction.
Main Differences Between Arithmetic Sequence and Linear Function in Points
- An arithmetic sequence may be expressed using a linear function if the function is of the linear type. On the other hand, the linear function can never be expressed in the form of an arithmetic series.
- Utilizing arithmetic sequence in a calculation may result in the production of a graph that is a slanted straight line. A linear function will always produce a graph on the horizontal plane that is perpendicular to one of the axes.
- The slope cannot be obtained directly from an arithmetic series; this is not feasible. On the other hand, a linear function does not provide us with an initial slope straight away.
- The slope of an arithmetic function could be seen from the graph. It is possible to calculate the slope of a linear function by using the following expression:
- On the other hand, linear functions that are linear are always continuous.
- The natural numbers or a subset of the natural numbers are the ones that are used to create arithmetic sequences, while the real numbers are the ones that are used to describe linear functions.
- The linear equation could have an infinite number of possible solutions, but the series can only have a finite number of possible outcomes at any one time.
- A sequence may be thought of as a function. However, sequences can only include values that are greater than zero. The notation "f(x) = x" can be used to describe functions; however, this cannot be done with sequences.
Conclusion
A lot of parallels may be seen between the liner function and the arithmetic sequence.
By making use of functional notation, we can get further pieces of information. In conclusion, the linear function is always a valid method for doing data analysis. Because the value of one function can be obtained by adding a specified amount to the value of the other function in a linear equation, it may be shown that the two functions are comparable to one another. Because the slope is made steeper, the slope that results from it is also made steeper.
Mathematical sequences suffer from the same issue, which occurs when a series is destined to either ascend or descend by the same predetermined amount.
References
- https://www.sciencedirect.com/science/article/pii/S0096300308008837
- https://arxiv.org/pdf/1403.0665
