Introduction

In terms of mathematics, a theorem is a set of statements about any specific topic that has been proved by providing various logical arguments, evidence, and reasons. Theorem statements can be demonstrated with help of mathematical operations. This article revolves around different theorems related with progression. Dirichlet’s theorem based on AP, theorem related with progression will be discussed in this article. This theorem mainly focuses on three types of progression namely arithmetic, geometric, and harmonic progression.

Mathematical explanation of theorem

A theorem can be stated as a general statement that is already proved or it can be proved using mathematical logical evidence and reasons. The theorem can be proved by providing logical arguments using different rules of the deductive system and logical consequences of already proved theorems or axioms. Theorems related to progression are proved using concepts of different progression.

Stages of theorem

In geometry, theorems are proved by following a specific structure. Following a particular structure, it provides a significant explanation while proving a theorem. Using steps and stages while proving a theorem, provides a systemic representation of what needs to be proved and what is required to prove a theorem. Stages of the theorem are:

• Theorem statement
• Figure
• Hypothesis
• Conclusion
• Construction
• Proof

Dirichlet’s theorem based on arithmetic progressions

The Dirichlet had provided a theorem related to progression which mainly focuses on arithmetic progression. This theorem is also known as the Dirichlet prime number theorem. According to this theorem, infinitely many prime numbers lie between two numbers. In this regard, if two positive co-prime numbers or integers “a” and “d” are taken then an infinite number of primes formulated using a + nd, provided that n is positive integer. Similarly, it is stated as there exist infinitely primes that are congruent to a modulo d. 

Numerical explanation can be represented by using two positive co-prime numbers a and d in arithmetic progression as: a, a + d, a + 2d, a + 3d. .

This theorem related with progression is proved by using Dirichlet L-function. Using some calculus and analytic number theory, this theorem was proved by scholars. 

How to find theorem related with progression

Theorems related with progression can be found by proving a theorem statement using already proved theorems or axioms. Theorem can be proved by following a systematic theorem structure and suitable diagram. Divergent and convergent theorems can be found using different mathematical operations such as using ∑ and natural logarithms and other basic mathematical operations. 

Theorems related with progression

Theorems related with progression contain theorems based on arithmetic progression, geometric progression, and harmonic progression. 

Some of theorems related with progression are mentioned below:

• A sequence a1, a2, a3,…, an,… is said to be an arithmetic sequence or progression if an + 1 = an + d, n ∈ N, in which a1 is first term and d is a common difference then sequence is called arithmetic progression.
• A sequence is said to be a geometric progression if ratio to any term is same as compared to consecutive terms. In that case, a common factor is known as common ratio. The nth term of a geometric progression is an = arn – 1. 
• Let it assume, lim (n→∞) an = L and lim (n→∞) bn = M and k is some constant. Then

 lim (n→∞) kan = k lim (n→∞) an = kL 

• Lim (n→∞) |an| = 0 if and only if lim(n→∞) an = 0. This theorem states that size of an is observed close to zero provided if and only if a gets close to zero.
• One of most important theorems related with progression is “If a sequence is bounded and monotonic then it converges.”
• Suppose that ∑ an and ∑ bn are convergent series, and c is a constant. Then 

1. ∑ can is convergent and ∑ can = c ∑ an
2. ∑ (an + bn) is convergent and ∑ (an + bn) = ∑ an + ∑ bn.

Conclusion

It is to be concluded that theorems are the true statements that are proved using mathematical operations and previously proved axioms. Proceeding to the proof of a theorem, some steps are followed that represent the theorem in a systematic way. Theorems related with progression namely Dirichlet’s theorem based on arithmetic progression states that infinitely prime number lies between two positive co-prime numbers. It requires calculus explanation and analytical explanation to prove Dirichlet’s theorem. Apart from these, it is found that different mathematical operations are requires proving theorem related with progression. Theorems related to arithmetic progression, geometric progression and harmonic progression have been proved through using natural logarithm and application of summations ∑.