Mathematicsis not just a branch of study. It is a part of our life. The techniques of maths are applied in our everyday lives. Generally the patterns of the occurrence OS found and then various generalizations are found. In most of the real-time situations calculus is involved.
The basis of this can be traced to the linear function. Some of the real-time samples are also based on the progressions and series. An arithmetic progression is formed by adding the same value to the previous number in order to obtain the next number. A linear function is used to establish a relationship with various other equations so as to obtain the solution.
Arithmetic Sequence vs Linear Function
The main difference between arithmetic sequence and linear function is that an arithmetic sequence is a sequence of numbers increasing or decreasing with a constant difference whereas a linear function is a polynomial function.
Comparison Table Between Arithmetic Sequence and Linear Function (in Tabular Form)
| Parameters | Linear Algebra | Arithmetic sequence |
|---|---|---|
| Branch of Maths | It is used in Calculus and Linear Algebra. | It is used in general mathematical calculations which are quite simple. |
| Values | Here constant values are obtained. | The constant values can not be obtained. |
| Plotting of Graph | Only a straight line will be obtained. | Here the graph can be plotted on both positive and negative sides. |
| Application | To find the area of space. | To count the number of things. |
| Area | When we calculate the area using the plot, we will obtain a constant area. | When the area is calculated, the area differs from one to the other. |
What is Arithmetic Sequence?
An arithmetic sequence is otherwise called as arithmetic progression. An arithmetic sequence is a list of numbers that has a common difference between the numbers. In an arithmetic sequence there will be a constant difference between the consecutive numbers. It is called a sequence because it follows a definite pattern throughout the sequence.
The constant difference that occurs between the two numbers is called a common difference. It is denoted by ‘d’. This common difference travels along the sequence. The common difference is used to travel from one number to the other. Because by adding or subtracting the common difference with the previous number we can obtain the preceding numbers in the sequence. In this way the whole sequence is generated.
When the difference between the consecutive terms is positive the sequence is said to be an increasing sequence. When the difference between the consecutive terms is negative sequence is said to be a decreasing sequence.
An arithmetic sequence which is finite in nature is called as finite arithmetic progression. An arithmetic series is the sum of the arithmetic progressi. Under the common difference, the arithmetic progression will behave. There are two types of infinity. An arithmetic progression may have either positive infinity or it can have a negative infinity.
- If the common difference is positive then the members of the sequence will reach positive infinity
- If the common difference is negative then the members of the sequence will reach negative infinity
We use the application of arithmetic sequence in our everyday life. For example, consider a roll of paper. Here the diameter of the role is considered to be the first term and the common difference is the twice the thickness of the paper. So using this we can find the entire roll. There are many other applications.
- The seats in a theatre are arranged in the arithmetic Progression method.
- The length of each rung in a ladder forms an arithmetic progression.
- The score in a basketball also forms an A.P as it keeps increasing by 1.
What is Linear Function?
The term Linear Function is now used in two areas of Mathematics. They are Calculus and Linear Algebra. In a Calculus, the linear function will be a straight graph. It is a polynomial function with a straight line graph and its degree may be one or Zero. The linear function is also used in mathematical analysis and functional analysis. Here the plot is a linear map.
In case if calculus or in analytical geometry, the Linear Function is a polynomial whose degree is either one or even less than one. The polynomials that have zero degrees are also included. When the degree of the polynomial is zero then that linear function becomes a constant function. When a graph s plotted for this constant function, a horizontal line is obtained.
In linear algebra, the linear function is used to obtain the area of a particular space. It is also used to establish a relationship between the two coordinates which will give rise to a third term. This application can be seen while plotting the speed, time, and distance graph.
Main Differences Between Arithmetic Sequence and Linear Function
- An arithmetic sequence can be represented as a linear function. But a linear function, can not be expressed as an arithmetic sequence
- In arithmetic sequence a straight line graph that is in a slanting manner is obtained. In a linear function, a horizontal graph parallel to one of the axis is obtained
- Slope can not be obtained from an arithmetic sequence directly. But linear function, does not give us a slope directly
- The slope in an arithmetic function can be obtained from the graph. But in a linear function, the slope can be found using the expression
- An arithmetic sequence is discrete but a linear function is continuous
Conclusion
The arithmetic sequence and the liner function re quite closely similar to each other.
We can more information using functional notation. Hence the linear function is always useful in obtaining the information from the data. Both are similar because in a linear equation when a certain amount is added to one of the functions then the value of the other function is created by similar value. Hence the slope obtained from it is also increased.
This is the same in the arithmetic sequence where a sequence gets increasing or decreasing by the same fixed value.
References
- https://www.sciencedirect.com/science/article/pii/S0096300308008837
- https://arxiv.org/pdf/1403.0665