--- aliases: [arithmetic sequence, arithmetic] --- #recursion-induction ## Definition > [!tldr] Definition > A [[sequence]] of numbers is said to be **arithmetic** if each number in the sequence has a common difference with the previous term. Put differently, a sequence is arithmetic if the difference of any two consecutive terms is constant. Notes: - We say that arithmetics sequences "grow at a constant rate". - Arithmetic sequences have a [[Recursion|recursive]] definition by letting $a_0$ be the first term of the sequence, then $a_n = b + a_{n-1}$ where $b$ is the common difference ## Examples and Non-Examples - The sequence $3, 5, 7, 9, 11, 13, \dots$ is arithmetic because each term is $2$ plus the previous term. (Or, the difference between any two consecutive terms is $2$.) - The sequence $10, 5, 0, -5, -10, -15, \dots$ is arithmetic because each item is $-5$ plus the previous term. - The sequence $3, 6, 12, 24, 48, \dots$ is *not* arithmetic because there is not a common difference between the terms. For example $6-3 = 3$ but $12 - 6 \neq 3$. (This is a [[Geometric sequence|geometric sequence]] with a common ratio of $2$.) ## Resources <div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/638676941?badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Screencast 5.3: Arithmetic and geometric sequences"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script> Other resources: - Tutorial: [Introduction to Arithmetic Sequences](https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-arithmetic-sequences/a/introduction-to-arithmetic-sequences) (Khan Academy) - Video: [How to find the nth term of an arithmetic sequence](https://www.youtube.com/watch?v=PStn9zHgXHU)