Sum of an Infinite Arithmetic Progression – Mindspark

Infinite Arithmetic Progression

An arithmetic progression is a series, where the difference between every two successive terms in the series is constant. For example, consider a series 7, 11, 15, 19, 23 and 27. In this series, the difference between every two successive terms is 4 and hence this is an arithmetic progression. This is a series containing 6 terms. If an arithmetic progression series has an infinite number of terms, it is called an infinite arithmetic progression. Let us learn how to calculate the sum of an infinite arithmetic progression here.

Sum of an infinite arithmetic progression:

Before learning the sum of an infinite AP, it is important to understand the concept of divergent and convergent.

A series is said to be a convergent series if its sum approaches a finite number. A good example of a convergent series is an infinite geometric progression where the |r| < 1 and r≠ 0 where r is the common ratio of the GP. 

A divergent series is an infinite series that is not convergent. An infinite series where the numbers do not approach zero is diverging. An infinite arithmetic progression is an example of a diverging series. 

In an infinite arithmetic progression where n is the number of terms, n → ∞ , and the common difference is greater than 0, the sum of the arithmetic progression would be infinitely large, and S∞ = ∞.

Similarly, in an infinite arithmetic progression where n → ∞ and has a common difference less than 0, then the terms of the Arithmetic Progression are approaching -∞ and the sum of such series would be S∞ = -∞.

For example, the sum of an infinite series having the terms -2, 0, 2, 4, 6……∞ will be ∞ since the common difference 2 is greater than 0. The sum of infinite series having the terms 7, 3, -1, -5 ….. -∞ will be -∞ since the common difference (-4) < 0.

Solved Examples:

1. If Mr. Y had Rs. 100 on Day 1 and every succeeding day, he received Rs. 20 more than what he received the previous day, for 15^{\text {th}}days. How much money does he have at the end of the 15^{\text {th}} day?

Solution:

From the given question, we can form a series having the terms 100, 120, 140, …, 380 where the first term a = 100, common difference d = 20, and the number of terms n = 15. We can calculate the total sum of amount Mr. Y has on Day 15 by using the formula:

S_{n}=\frac{n}{2}[2 a+(n-1) d]

Substituting the given figures in the above formula, we have:

S_{15}=\frac{15}{2}[2(100)+(15-1) 20]

       =\frac{15}{2}[200+(14) 20]

       =\frac{15}{2}[200+280]

       =\frac{15}{2}\times 480

       =15\times 240

       =3600

\therefore S_{15}=3600

Hence, the total amount Mr. Y had on Day 15 was Rs. 3600.

2. Find the sum of an infinite AP having the terms 1, 5, 9, 13, …., ∞.

Since infinite arithmetic progression series are divergent series, the sum of an infinite arithmetic series can not exactly be determined. The common difference in this AP is 4 and so we know that the sum of the given AP is ∞.