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Questions: Algebra BusinessCalculus
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Find the rational number (fractional) representation of the repeating decimal \(\displaystyle 0.610610610\ldots\)
The first term is \(\displaystyle a_1=0.610=\frac{610}{1000}=\frac{61}{100}\). 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 \(\displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{610}{1-\left(\frac{1}{1000}\right)}=\frac{610}{999}\).
\begin{question}Find the rational number (fractional) representation of the repeating decimal $0.610610610\ldots$
\soln{9cm}{The first term is $a_1=0.610=\frac{610}{1000}=\frac{61}{100}$. 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 $S_{\infty}=\frac{a_1}{1-r}=\frac{610}{1-\left(\frac{1}{1000}\right)}=\frac{610}{999}$. }
\end{question}
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\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 0.610610610\ldots " src="/equation_images/%20%5Cdisplaystyle%200.610610610%5Cldots%20" alt="LaTeX: \displaystyle 0.610610610\ldots " data-equation-content=" \displaystyle 0.610610610\ldots " /> </p> </p>
<p> <p>The first term is <img class="equation_image" title=" \displaystyle a_1=0.610=\frac{610}{1000}=\frac{61}{100} " src="/equation_images/%20%5Cdisplaystyle%20a_1%3D0.610%3D%5Cfrac%7B610%7D%7B1000%7D%3D%5Cfrac%7B61%7D%7B100%7D%20" alt="LaTeX: \displaystyle a_1=0.610=\frac{610}{1000}=\frac{61}{100} " data-equation-content=" \displaystyle a_1=0.610=\frac{610}{1000}=\frac{61}{100} " /> . 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{610}{1-\left(\frac{1}{1000}\right)}=\frac{610}{999} " src="/equation_images/%20%5Cdisplaystyle%20S_%7B%5Cinfty%7D%3D%5Cfrac%7Ba_1%7D%7B1-r%7D%3D%5Cfrac%7B610%7D%7B1-%5Cleft%28%5Cfrac%7B1%7D%7B1000%7D%5Cright%29%7D%3D%5Cfrac%7B610%7D%7B999%7D%20" alt="LaTeX: \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{610}{1-\left(\frac{1}{1000}\right)}=\frac{610}{999} " data-equation-content=" \displaystyle S_{\infty}=\frac{a_1}{1-r}=\frac{610}{1-\left(\frac{1}{1000}\right)}=\frac{610}{999} " /> . </p> </p>