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Introduction
The sum of arithmetic series calculates the summation of nth numbers of terms in an arithmetic progression, which is called finite arithmetic series. A sum of infinite arithmetic series calculates the summation of infinite numbers. The arithmetic series has been found useful in India to solve real-life problems associated with certain patterns. By the end of this study, the significance of the sum of finite arithmetic series and the sum of infinite arithmetic series are going to be discussed.
Concept of the Sum of Arithmetic Series
The sum of the arithmetic series also termed ‘the sum of finite arithmetic series’ is used to calculate the summation of successive terms in an arithmetic progression (AP). The series includes a sequence of numbers in which each successive number is the summation of preceding terms, which is a fixed number. This fixed number is known as the common difference and therefore, the difference between every two consecutive terms in an arithmetic series is the same. In ancient India, the sum of the arithmetic series was important because this helped to identify numerical associated with patterns.
Defining the formula of Sum of Arithmetic Series
The formula of the sum of arithmetic series refers to the calculation of the sum of each term present in the arithmetic sequence. The addition method of the series follows the sum of nth terms that is the sum of a finite number of terms. The terms of the series can be represented by (m, m+d, m+2d, m+3d, ……, Nth) where ‘m’ is the first term in the series and ‘d’ is a common difference.
The general formula for the sum of an arithmetic series is given below:
Formula 1: Sum (nth term) = nth term/2[2m+(nth term – 1) d]
Formula 2: sum (nth term) = nthterm/2[m (first term) + m (nth term)]
Here,
Sum (nth term) = the summation of the nth numbers of arithmetic series
m (first term) = the first term in the series
d= common difference between two consecutive terms
nth term= total number of terms in the series
m (nth term) = the last term in the series
What is understood by the sum of finite arithmetic series?
The sum of finite arithmetic series consists of a sequence of numbers with a constant difference that exists between the preceding term and the succeeding term. An example has been elaborated below:
{2, 4, 6, 8, 10, …….., 50} is a finite arithmetic series.
The general formula to determine the sum is nth term/2[2m+(nth term – 1) d].
What is meant by the sum of infinite arithmetic series?
The sum of the infinite arithmetic series calculates the total of the sequence of numbers. However, in this type of arithmetic progression, the number of terms is infinite or undefined. An example has been elaborated of an infinite sequence of terms,
1/2, 1/4, 1/8, 1/16, ……. = T
Where T is the sum of the infinite arithmetic series
Convergent Infinite Series
In convergent infinite series, a limit exists and the infinite series converges. For example, 1/2, 1/4, 1/8, 1/16, …….
Term sum
1/2 0.50
1/4 0.750
1/8 0.8750
1/16 0.93750
1/32 0.96870
Therefore, the summation of the above-mentioned series is approximately 1, hence it could be said that the sum of the infinite series is convergent.
Divergent Infinite Series
In divergent infinite series, the limit does not exist or it could be said that the limit is infinite thus the series is diverging. For example, 2 – 2 + 2 – 2 + 2 ……
The summation of the above-mentioned series does not result in the value; hence, it could be said that the sum of the infinite series is divergent.
Difference between the sum of finite arithmetic series and the sum of infinite arithmetic series
In the sum of finite arithmetic series, a limit exists in the sequence of the number terms. However, in the infinite arithmetic series, there is no limit that exists in the number sequences.
The number of terms in the sum of finite arithmetic series can be counted and summed up, while in the sum of infinite arithmetic series the sequence of the numbers cannot be counted or summed up.
Conclusion
The discussion on the sum of the arithmetic series, the sum of the finite arithmetic series, and the sum of the infinite arithmetic series has been analyzed. It can be concluded that the arithmetic sequences are important to solve mathematical as well as many real-life applications. In addition, this is also useful in the areas such as statistics, physics, and finance to identify and recognize patterns.
