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The common ratio is \(\displaystyle r=\frac{a_2}{a_1}=-12\). Using the first term and the common ratio gives \(\displaystyle a_n=6\left( -12 \right)^{n-1}\)

Download \(\LaTeX\)

\begin{question}Find the general term of the sequence $6,-72,864,\ldots$
    \soln{9cm}{The common ratio is $r=\frac{a_2}{a_1}=-12$. Using the first term and the common ratio gives $a_n=6\left( -12 \right)^{n-1}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
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\newcommand{\hideifShowSolution}[1]{%
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%
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\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

Find the general term of the sequence <img class="equation_image" title=" \displaystyle 6,-72,864,\ldots " src="/equation_images/%20%5Cdisplaystyle%206%2C-72%2C864%2C%5Cldots%20" alt="LaTeX: \displaystyle 6,-72,864,\ldots " data-equation-content=" \displaystyle 6,-72,864,\ldots " />

The common ratio is <img class="equation_image" title=" \displaystyle r=\frac{a_2}{a_1}=-12 " src="/equation_images/%20%5Cdisplaystyle%20r%3D%5Cfrac%7Ba_2%7D%7Ba_1%7D%3D-12%20" alt="LaTeX: \displaystyle r=\frac{a_2}{a_1}=-12 " data-equation-content=" \displaystyle r=\frac{a_2}{a_1}=-12 " /> . Using the first term and the common ratio gives <img class="equation_image" title=" \displaystyle a_n=6\left( -12 \right)^{n-1} " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D6%5Cleft%28%20-12%20%5Cright%29%5E%7Bn-1%7D%20" alt="LaTeX: \displaystyle a_n=6\left( -12 \right)^{n-1} " data-equation-content=" \displaystyle a_n=6\left( -12 \right)^{n-1} " />