An **arithmetic progression** (or arithmetic sequence) is a list of numbers, where every term increases by the same amount—called a common difference. In other words, to go from one term to the next, you just add a number. For example, starting with 1:

- {1, 2, 3} … add 1 each time,
- {1, 7, 14} … add 7 each time.

You don’t have to start with 1 though; You can start with any number. What’s important is that you add the same constant each time.

## Finding Common Difference in Arithmetic Progression

The common difference is how much is added to each term in the sequence. For example, the sequence {2, 4, 6, 8} increases by 2—the common difference. Another example: {1, 25, 49} has a common difference of 24. In notation, the common difference is often written as *d*. For example:

a, a + d, a + 2d, a + 3d, a + 4d.

Which can also be written as a + (n · d), where n starts at zero.

For example, if a = 1 and d = 2:

1, 1 + 2, 1 + 2(2), 1 + 3(2), 1 + 4(2) = 1, 2, 5, 7, 9.

**Example question: **What is the common difference for the arithmetic progression {5, 8, 11, 14, 17}?

Step 1: Identify the first term. In this list, that’s 5.

Step 2: Subtract the first term from the second: 8 – 5 = 3.

The common difference is 3.

## Finite and Infinite Arithmetic Progression

The progression can be a fixed (finite) amount of numbers or an infinite amount. For example, starting with 2 and using a common difference of 3, you get the arithmetic sequence {2, 5, 8, 11…}. The three dots (…) indicate that the sequence goes on an on until infinity. When you have a fixed number of terms, the sequence is called an n-term arithmetic progression. For example, the sequence {2, 5, 8} is a three term arithmetic sequence.

## Arithmetic Progression Examples from Number Theory

Arithmetic progression is heavily used in number theory—especially in the analysis of prime numbers. A rather more complex example of an arithmetic progression from number theory:

**{a + mk: k >0} ⊆**

Where:

- m = fixed integer > 0 &
- a = fixed integer ≥ 0
- Where:
- a + mk = equals a prime number,
- gcd(a, m) = 1.

- Where:
- ⊆ = subset of (or equal to),
- k = a natural number,
- ℕ = set of natural numbers.

The above example is called **Dirichlet’s theorem on primes in arithmetic progressions** (Lozano-Robledo, 2019). Related to Dirichlet’s theorem are these two important examples of arithmetic progression, which contain all prime numbers except for 2 and 3 (Caldwell, 2020):

- 1, 7, 13, 19, 25, 31, 37, …
- 5, 11, 17, 23, 29, 35, 41, …

## References

Caldwell, C. (2020). Arithmetic Sequence. Retrieved August 26, 2020 from: https://primes.utm.edu/glossary/page.php?sort=ArithmeticSequence

Lozano-Robledo, A. (2019). Number Theory and Geometry. An Introduction to Arithmetic Geometry. American Mathematical Society.

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