7.6855172413793103 in fractions is?
1. Continued fraction of 7.6855172413793103 :
$$\small{\text{$
\begin{array}{rrrrrrrrrr}
[x_0;&x_1,&x_2,&x_3,&x_4,&x_5,&x_6,&x_7,&x_8,&x_9]\\
\[[7;&1,&2,&5,&1,&1,&3,&1,&1,&2]
\end{array}
$
}}$$
$$\small{\text{$
\begin{array}{lclcl}
\textcolor[rgb]{1,0,0}{x_0} &=& \textcolor[rgb]{1,0,0}{7}.6855172413793103 \\
&& 1/0.6855172413793103= \\
\textcolor[rgb]{1,0,0}{x_1}&=&\textcolor[rgb]{1,0,0}{1}.4587525150905433549486051115546459473433926066921394\\
&& 1/0.4587525150905433549486051115546459473433926066921394=\\
\textcolor[rgb]{1,0,0}{x_2}&=&\textcolor[rgb]{1,0,0}{2}.1798245614035083186653585718683317691923088345579715\\
&& 1/0.1798245614035083186653585718683317691923088345579715=\\
\textcolor[rgb]{1,0,0}{x_3}&=&\textcolor[rgb]{1,0,0}{5}.5609756097561115779298036883141182477038929421104669\\
&& 1/0.5609756097561115779298036883141182477038929421104669=\\
\textcolor[rgb]{1,0,0}{x_4}&=&\textcolor[rgb]{1,0,0}{1}.7826086956521293714555765605533410043560222722903369\\
&& 1/0.7826086956521293714555765605533410043560222722903369=\\
\textcolor[rgb]{1,0,0}{x_5}&=&\textcolor[rgb]{1,0,0}{1}.2777777777778505015432098790380390517833503327285810\\
&& 1/0.2777777777778505015432098790380390517833503327285810=\\
\textcolor[rgb]{1,0,0}{x_6}&=&\textcolor[rgb]{1,0,0}{3}.5999999999990575000000002144187499999512197343750242\\
&& 1/0.5999999999990575000000002144187499999512197343750242=\\
\textcolor[rgb]{1,0,0}{x_7}&=&\textcolor[rgb]{1,0,0}{1}.6666666666692847222222257391435185232429161265495199\\
&& 1/0.6666666666692847222222257391435185232429161265495199=\\
\textcolor[rgb]{1,0,0}{x_8}&=&\textcolor[rgb]{1,0,0}{1}.4999999999941093750000152199023437106755773194376239\\
&& 1/0.4999999999941093750000152199023437106755773194376239=\\
\textcolor[rgb]{1,0,0}{x_9}&=&\textcolor[rgb]{1,0,0}{2}.0000000000\ldots
\end{array}
$}}$$
2. The successive convergents with numerators p and denominators q are given by the
$$\small{ \text{ Formula }
\boxed{
\begin{array}{lcl}
\dfrac{p_{n+1}}{q_{n+1}} = \dfrac{x_{n+1}\cdot p_n + p_{n-1} }{x_{n+1}\cdot q_n + q_{n-1} } \qquad \dfrac{p_0} {q_0} = \dfrac{x_0}{1}=\dfrac{ \textcolor[rgb]{1,0,0}{7} }{1} \qquad \dfrac{p_{-1}} {q_{-1} } = \dfrac{1}{0} \\\\
$ For example $ n=0:~~
\dfrac{p_{1}}{q_{1}}
= \dfrac{x_1\cdot p_0 + p_{-1} }{x_1\cdot q_0 + q_{-1} }
= \dfrac{x_1\cdot x_0+1}{x_1\cdot 1+ 0 }
= \dfrac{\textcolor[rgb]{1,0,0}{1}\cdot 7 + 1 }{ \textcolor[rgb]{1,0,0}{1}\cdot 1 + 0} = \dfrac{8}{1} = 8\\\\
$ For example $ n=1:~~
\dfrac{p_{2}}{q_{2}}
= \dfrac{x_2\cdot p_1 + p_0 }{x_2\cdot q_1 + q_0 }
= \dfrac{x_2\cdot 8 + 7}{x_2\cdot 1+ 1 }
= \dfrac{\textcolor[rgb]{1,0,0}{2}\cdot 8 + 7 }{ \textcolor[rgb]{1,0,0}{2}\cdot 1+ 1} = \dfrac{23}{3} = 7.6\overline{6}\\\\
$ For example $ n=2:~~
\dfrac{p_{3}}{q_{3}}
= \dfrac{x_3\cdot p_2 + p_1 }{x_3\cdot q_2 + q_1 }
= \dfrac{x_3\cdot 23 + 8}{x_3\cdot 3+ 1 }
= \dfrac{\textcolor[rgb]{1,0,0}{5}\cdot 23 + 8 }{ \textcolor[rgb]{1,0,0}{5}\cdot 3+ 1} = \dfrac{123}{16} = 7.6875\\\\
$ For example $ n=3:~~
\dfrac{p_{4}}{q_{4}}
= \dfrac{x_4\cdot p_3 + p_2 }{x_4\cdot q_3 + q_2 }
= \dfrac{x_4\cdot 123 + 23}{x_4\cdot 16 + 3 }
= \dfrac{\textcolor[rgb]{1,0,0}{1}\cdot 123 + 23 }{ \textcolor[rgb]{1,0,0}{1}\cdot 16 + 3 } = \dfrac{146}{19} = 7.68421052631578947\ldots \\\\
$ For example $ n=4:~~
\dfrac{p_{5}}{q_{5}}
= \dfrac{x_5\cdot p_4 + p_3 }{x_5\cdot q_4 + q_3 }
= \dfrac{x_5\cdot 146 +123}{x_5\cdot 19+16 }
= \dfrac{\textcolor[rgb]{1,0,0}{1}\cdot 146 +123 }{ \textcolor[rgb]{1,0,0}{1}\cdot 19+16 } = \dfrac{269}{35} = 7.68571428571428571\ldots \\\\
$ For example $ n=5:~~
\dfrac{p_{6}}{q_{6}}
= \dfrac{x_6\cdot p_5 + p_4 }{x_6\cdot q_5 + q_4 }
= \dfrac{x_6\cdot 269+ 146}{x_6\cdot 35+19 }
= \dfrac{\textcolor[rgb]{1,0,0}{3}\cdot 269+ 146 }{ \textcolor[rgb]{1,0,0}{3}\cdot 35+19 } = \dfrac{953}{124} = 7.68548387096774194\ldots \\\\
$ For example $ n=6:~~
\dfrac{p_{7}}{q_{7}}
= \dfrac{x_7\cdot p_6 + p_5 }{x_7\cdot q_6 + q_5 }
= \dfrac{x_7\cdot 953+ 269 }{x_7\cdot 124 + 35 }
= \dfrac{\textcolor[rgb]{1,0,0}{1}\cdot 953+ 269 }{ \textcolor[rgb]{1,0,0}{1}\cdot 124 + 35 } = \dfrac{1222}{159} = 7.68553459119496855\ldots \\\\
$ For example $ n=7:~~
\dfrac{p_{8}}{q_{8}}
= \dfrac{x_8\cdot p_7 + p_6 }{x_8\cdot q_7 + q_6 }
= \dfrac{x_8\cdot 1222 + 953 }{x_8\cdot 159 + 124 }
= \dfrac{\textcolor[rgb]{1,0,0}{1}\cdot 1222 + 953 }{ \textcolor[rgb]{1,0,0}{1}\cdot 159 + 124 } = \dfrac{2175}{283} = 7.68551236749116607\ldots \\\\
$ For example $ n=8:~~
\dfrac{p_{9}}{q_{9}}
= \dfrac{x_9\cdot p_8 + p_7 }{x_9\cdot q_8 + q_7 }
= \dfrac{x_9\cdot 2175 + 1222 }{x_9\cdot 283 + 159 }
= \dfrac{\textcolor[rgb]{1,0,0}{2}\cdot 2175 + 1222 }{ \textcolor[rgb]{1,0,0}{2}\cdot 283 + 159 } = \dfrac{5572}{725} = 7.68551724137931034\ldots \\\\
\end{array}
}
}$$
$$7.6855172413793103 \approx \dfrac{5572}{725} = 7.68551724137931034\ldots$$
+26400 
+118723 
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