To find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$, we can set up a long division to find the decimal representation of $\frac{1}{17}$.
$\begin{array}{r|llllllllllllllllllllllllllllll}
\multicolumn{2}{r}{0.\underline{0588235294117647}} \\
17 & 1.0000000000000000 \\
\multicolumn{2}{r}{85} \\
\cline{2-2}
\multicolumn{2}{r}{20} \\
\multicolumn{2}{r}{17} \\
\cline{2-2}
\multicolumn{2}{r}{30} \\
\multicolumn{2}{r}{17} \\
\cline{2-2}
\multicolumn{2}{r}{130} \\
\multicolumn{2}{r}{119} \\
\cline{2-2}
\multicolumn{2}{r}{110} \\
\multicolumn{2}{r}{102} \\
\cline{2-2}
\multicolumn{2}{r}{800} \\
\multicolumn{2}{r}{748} \\
\cline{2-2}
\multicolumn{2}{r}{520} \\
\multicolumn{2}{r}{476} \\
\multicolumn{2}{r}{85} \\
\cline{2-3}
\multicolumn{2}{r}{440} \\
\multicolumn{2}{r}{408} \\
\cline{2-2}
\multicolumn{2}{r}{320} \\
\multicolumn{2}{r}{289} \\
\cline{2-3}
\multicolumn{2}{r}{3100} \\
\multicolumn{2}{r}{3063} \\
\cline{2-2}
\multicolumn{2}{r}{3700} \\
\multicolumn{2}{r}{3571} \\
\cline{2-2}
\multicolumn{2}{r}{1290} \\
\multicolumn{2}{r}{1190} \\
\cline{2-2}
\multicolumn{2}{r}{1000} \\
\multicolumn{2}{r}{986} \\
\cline{2-2}
\multicolumn{2}{r}{1400} \\
\multicolumn{2}{r}{1190} \\
\cline{2-2}
\multicolumn{2}{r}{2100} \\
\multicolumn{2}{r}{2048} \\
\cline{2-2}
\multicolumn{2}{r}{5200} \\
\end{array}$
After a few more iterations, we find that the decimal representation of $\frac{1}{17}$ is $0.\overline{0588235294117647}$. Therefore, the $4000$th digit following the decimal point is $\boxed{4}$.
Find the $4000$th digit following the decimal point in the expansion of $\frac{1}{17}$
1 answer