\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the rational number (fractional) representation of the repeating decimal \(\displaystyle 0.306306306\ldots\)
The first term is \(\displaystyle a_1=0.306=\frac{306}{1000}=\frac{153}{500}\). The length of the repeating block is 3, so the common ratio is \(\displaystyle \frac{1}{1000}\). Using the formula for the sum of an infinite geometric series gives \(\displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{306}{1-\left(\frac{1}{1000}\right)}=\frac{34}{111}\).
\begin{question}Find the rational number (fractional) representation of the repeating decimal $0.306306306\ldots$
\soln{9cm}{The first term is $a_1=0.306=\frac{306}{1000}=\frac{153}{500}$. The length of the repeating block is 3, so the common ratio is $\frac{1}{1000}$. Using the formula for the sum of an infinite geometric series gives $S_{\infty}=\frac{a_1}{1-r}=\frac{306}{1-\left(\frac{1}{1000}\right)}=\frac{34}{111}$. }
\end{question}
\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%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the rational number (fractional) representation of the repeating decimal <img class="equation_image" title=" \displaystyle 0.306306306\ldots " src="/equation_images/%20%5Cdisplaystyle%200.306306306%5Cldots%20" alt="LaTeX: \displaystyle 0.306306306\ldots " data-equation-content=" \displaystyle 0.306306306\ldots " /> </p> </p>
<p> <p>The first term is <img class="equation_image" title=" \displaystyle a_1=0.306=\frac{306}{1000}=\frac{153}{500} " src="/equation_images/%20%5Cdisplaystyle%20a_1%3D0.306%3D%5Cfrac%7B306%7D%7B1000%7D%3D%5Cfrac%7B153%7D%7B500%7D%20" alt="LaTeX: \displaystyle a_1=0.306=\frac{306}{1000}=\frac{153}{500} " data-equation-content=" \displaystyle a_1=0.306=\frac{306}{1000}=\frac{153}{500} " /> . The length of the repeating block is 3, so the common ratio is <img class="equation_image" title=" \displaystyle \frac{1}{1000} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B1000%7D%20" alt="LaTeX: \displaystyle \frac{1}{1000} " data-equation-content=" \displaystyle \frac{1}{1000} " /> . Using the formula for the sum of an infinite geometric series gives <img class="equation_image" title=" \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{306}{1-\left(\frac{1}{1000}\right)}=\frac{34}{111} " src="/equation_images/%20%5Cdisplaystyle%20S_%7B%5Cinfty%7D%3D%5Cfrac%7Ba_1%7D%7B1-r%7D%3D%5Cfrac%7B306%7D%7B1-%5Cleft%28%5Cfrac%7B1%7D%7B1000%7D%5Cright%29%7D%3D%5Cfrac%7B34%7D%7B111%7D%20" alt="LaTeX: \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{306}{1-\left(\frac{1}{1000}\right)}=\frac{34}{111} " data-equation-content=" \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{306}{1-\left(\frac{1}{1000}\right)}=\frac{34}{111} " /> . </p> </p>