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The common ratio is \(\displaystyle r=\frac{a_2}{a_1}=-5\). Using the first term and the common ratio gives \(\displaystyle a_n=5\left( -5 \right)^{n-1}\)

Download \(\LaTeX\)

\begin{question}Find the general term of the sequence $5,-25,125,\ldots$
    \soln{9cm}{The common ratio is $r=\frac{a_2}{a_1}=-5$. Using the first term and the common ratio gives $a_n=5\left( -5 \right)^{n-1}$}

\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}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
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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 5,-25,125,\ldots " src="/equation_images/%20%5Cdisplaystyle%205%2C-25%2C125%2C%5Cldots%20" alt="LaTeX: \displaystyle 5,-25,125,\ldots " data-equation-content=" \displaystyle 5,-25,125,\ldots " />

The common ratio is <img class="equation_image" title=" \displaystyle r=\frac{a_2}{a_1}=-5 " src="/equation_images/%20%5Cdisplaystyle%20r%3D%5Cfrac%7Ba_2%7D%7Ba_1%7D%3D-5%20" alt="LaTeX: \displaystyle r=\frac{a_2}{a_1}=-5 " data-equation-content=" \displaystyle r=\frac{a_2}{a_1}=-5 " /> . Using the first term and the common ratio gives <img class="equation_image" title=" \displaystyle a_n=5\left( -5 \right)^{n-1} " src="/equation_images/%20%5Cdisplaystyle%20a_n%3D5%5Cleft%28%20-5%20%5Cright%29%5E%7Bn-1%7D%20" alt="LaTeX: \displaystyle a_n=5\left( -5 \right)^{n-1} " data-equation-content=" \displaystyle a_n=5\left( -5 \right)^{n-1} " />