# Sequences A **sequence** of real numbers is an infinite string of numbers in a given order. $\{a_n\}^\infty_{n=1}=\{a_1, a_2, a_3,...,a_n,a_{n+1},...\}$ - $n$ is the *index variable*. - Each number $a_n$ is a *term* of the sequence. Sequences can sometimes be defined by **explicit formulas**, in which case: - $a_n = f(n)$ for some function $f(n)$ defined over the positive integers. They can also be represented using a [[recursion|recurrence]] relation, meaning that each term in the sequence is based on previous terms, not just $n$: - Ex: $a_n = 4a_{n-1}-3$; $a_1 = 3$ ## Properties of Sequences Finite: Can be counted and totaled (has an end) Infinite: Cannot be counted and totaled (does not have an end) - The [[Limits|limit as the sequence/series approaches infinity]] can be evaluated instead, A sequence is **monotonic** (aka “unchanging”) if it is non-increasing or non-decreasing. - A sequence ${a_n}$ is *non-decreasing* if $a_n \le a_{n+1}$ for all $n$. - It is *non-increasing* if $a_n \ge a_{n+1}$ for all $n$. ### The Limit of a Sequence The limit of a sequence $a_n$ when $n$ becomes a large number Convergence: ![[Pasted image 20240409163330.png|500]] ### Bounded Sequences ## Notable Sequences > See also: > - [[Sums and Series#Notable Series]] ![[Pasted image 20250613135218.png|400]] In an **arithmetic sequence**, the *difference* between every pair of consecutive terms is the same. > [!definition] Arithmetic Sequence > Contentsa In a **geometric sequence**, the *ratio* of every pair of consecutive terms is the same. The