How does one express $0.0\overline{1410}$ as a fraction? I know the answer is $141/9999$ but I am not sure how to derive it using some formula from below.
Wikipedia says assume $x=0.a_1a_2...a_n.$
$10^nx=a_1a_2...a_n.\overline{a_1a_2...a_n}$
$(10^{n}-1)x=99...99x=a_1a_2...a_n$
$x=\dfrac{a_1a_2..._n}{10^n-1}=\dfrac{a_1a_2...a_n}{99...99}$
What to do when $a_4=0$?
In our case where $n=4$, the length of the repetend.
$10^{n}x=141.0\overline{1410}.$
$(10^{n}-1)x=141$
$9999x=141$
$x=141/9999$
Generally, here's the process you can take. Say you have some decimal with $n$ non-repeating digits followed by $m$ repeating digits:
$$ 0.x_1 \cdots x_n\overline{y_1 \cdots y_m} . $$
Then, you can split this into a sum of two decimals:
$$ 0.x_1 \cdots x_n\overline{y_1 \cdots y_m} = 0.x_1 \cdots x_n + 0.\underbrace{0 \cdots 0}_{n \ 0's} \ \overline{y_1 \cdots y_m} . $$
Then, the first decimal is
$$ 0.x_1 \cdots x_n = \frac{x_1 \cdots x_n}{10^n} , $$
and the second decimal is
$$ 0.\underbrace{0 \cdots 0}_{n \ 0's} \ \overline{y_1 \cdots y_m} = \frac{1}{10^n}\cdot 0.\overline{y_1 \cdots y_m} . $$
You can use the same logic you quoted from Wikipedia in the question above to show that
$$ 0.\overline{y_1 \cdots y_m} = \frac{y_1 \cdots y_m}{\underbrace{9 \cdots 9}_{m \ 9's}} = \frac{y_1 \cdots y_m}{10^m - 1} . $$
Then, our second decimal from the sum above is
$$ 0.\underbrace{0 \cdots 0}_{n \ 0's} \ \overline{y_1 \cdots y_m} = \frac{y_1 \cdots y_m}{10^n\left(10^m-1\right)} . $$
Summing these fractions, we have
$$ 0.x_1 \cdots x_n\overline{y_1 \cdots y_m} = \frac{x_1 \cdots x_n}{10^n} + \frac{y_1 \cdots y_m}{10^n\left(10^m-1\right)} , $$
or in terms of the digits, this is
$$ 0.x_1 \cdots x_n\overline{y_1 \cdots y_m} = \frac{x_1 \cdots x_n}{1\underbrace{0 \cdots 0}_{n \ 0's}} + \frac{y_1 \cdots y_m}{\underbrace{9 \cdots 9}_{m \ 9's}\underbrace{0 \cdots 0}_{n \ 0's}} .$$