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week6/17-02-2022/17-02-2022.tex

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+%        File: 20-01-2022.tex
+%     Created: 12:28:35 Thu, 20 Jan 2022 EST
+% Last Change: 12:28:35 Thu, 20 Jan 2022 EST
+%
+\documentclass[letterpaper]{article}
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{cancel}
+\usepackage{amssymb}
+\usepackage{listings}
+\usepackage[shortlabels]{enumitem}
+\usepackage{soul}
+%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
+\usepackage{geometry}
+\usepackage{dirtytalk}
+\usepackage{mathtools}
+\usepackage{lplfitch}
+
+\geometry{portrait, margin=1in}
+
+%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
+
+\date{02/17/2022}
+\title{%
+        Sequences\\
+            \large MATH--190--01 : Discrete Mathematics for Computing}
+\author{Blizzard MacDougall}
+\begin{document}
+\maketitle
+\pagenumbering{arabic}
+Sequences can have duplicates.
+
+A sequence is a function whose domain is either all integers, restricted to either
+some range, or above some given integer. Sequences don't count down, we'd just write
+a negative function for it then.
+
+A sequence is typically denoted as $a_m, a_{m+1}, a_{m+2},\dots,a_n$, where $a_m$ is
+the initial term, and $a_n$ is the final term. $a_n$ need not exist, but the ellipses
+still are.
+
+Index superiority is stupid.
+
+An explicit formula (or general formula) is a rule to show how to find $a_k$, given $k$.
+
+Alternating sequences are very specific. Alternating specifically means swapping
+from positive to negative, not just oscillating between two values.
+
+A geometric progression is a sequence of form $a, ar, ar^2, ar^3,\dots$ This also
+brings in the idea of geometric mean. This is found by multiplying all values,
+and taking the $n$th root. We won't be using it much here.
+
+An arithmetic progression is a sequence of form $a, a+d, a+2d, a+3d, a+4d,\dots$
+
+A recurrence relation is an equation that bases off of its previous terms. A solution
+is some sequence that satisfies a recurrence relation. The Fibonacci sequence is
+a recurrence relation. A solution is found when we can remove the recursion from
+the sequence's definition. This is called a closed formula.
+
+You already know summation exists; double summation exists, its just a nested for loop.
+A summation can also be defined with a set under the summation sign.
+\end{document}