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Arithmetic Progression / Sequence

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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.
  • ⊆ = 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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Stephanie Glen. "Arithmetic Progression / Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/arithmetic-progression/
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