Sum of AP Formula for an Infinite AP
As we already know that AP are sequences that go up to infinity but finding the sum of AP up to an infinite term is a tedious task. If the AP is increasing AP the sum of the infinite terms of the AP is positive infinite but if the AP is decreasing AP then its sum of infinite terms is negative infinity. Before learning about them in detail let’s first learn what is Increasing AP, and what is decreasing AP.
Increasing AP
An AP where the next consecutive term is greater than the previous term is called an increasing AP. The common difference is always positive in increasing AP. (i.e. d > 0). Some examples of increasing AP are,
- S(n) = 2, 6, 10, 14,…
- S(n) = 5, 8, 11, 14,…
- S(n) = 6, 11, 16, 21,…
Sum of Infinite Terms of increasing AP = ∞
This can be proved as, suppose we have to find the sum of S(n) = 2, 6, 10, 14,… up to infinite terms, the formula for the sum of n-terms is,
Sn = n/2(2a + (n-1)d)
Putting a = 2 and n = ∞ we get,
Sn = ∞/2[2(2) + (∞-1)d]
Sn = ∞
Decreasing AP
An AP where the next consecutive term is lesser than the previous term is called a decreasing AP. The common difference is always negative in decreasing AP. (i.e. d < 0). Some examples of decreasing AP are,
- S(n) = 8, 4, 0, -4,…
- S(n) = -2, -4, -6, -8,…
- S(n) = 10, 7, 4, 1,…
Sum of Infinite Terms of decreasing AP = -∞
This can be proved as, suppose we have to find the sum of S(n) = 8, 4, 0, -4,… up to infinite terms, the formula for the sum of n-terms is,
Sn = n/2(2a + (n-1)d)
Putting a = 8 and n = ∞ we get,
Sn = ∞/2[2(8) + (∞-1)d]
As we know that if d is negative here.
Sn = -∞
We can summarize these conditions as,
- Sum of infinite AP = ∞, if d > 0
- Sum of infinite AP = -∞, if d > 0
Sum of N Terms of an AP
Arithmetic Sequence is defined as the sequence of numbers such that the difference between any two consecutive numbers is always constant. That is we can say that the next number in the arithmetic sequence is always the previous sequence plus the constant term, where the constant term can be both positive or negative. We can easily find the sum of the first “n” terms of the arithmetic sequence using the formula discussed in the article.
Johann Carl Friedrich Gauss in the 19th century was the first to find the sum of the first “n” terms of the arithmetic sequence where n is any natural number. Let us learn all about the sum of n terms of an AP along with examples in this article.
Table of Content
- What is Sum of n Terms of an AP?
- Sum of n Terms of AP Formula
- Sum of n Terms of AP Proof
- Sum of AP Formula for an Infinite AP
- Sum of AP Formulas
- Sum of Natural Numbers