The problem is equivalent to representing
$$\frac{1}{7} = 0.142857142857142857\dots := 0.\overline{142857}$$
as the sum of a finite number, $n$, of real numbers that only have digits $0$ and $7$ in their decimal representation. In order to get the fourth decimal $8$, it is easy to see that we need $n \geq 8$. An example with $n = 8$ terms is
$$\begin{aligned} 1 = &\ 0.\overline{777777} + 0.\overline{077777} + 0.\overline{070777} + 0.\overline{070777} \\ & + 0.\overline{000777} + 0.\overline{000707} + 0.\overline{000707} + 0.\overline{000700}, \end{aligned}$$
which leads to the following representation of $1$:
$$1 = \frac{777777 + 77777 + 70777 + 70777 + 777 + 707 + 707 + 700}{10^6 - 1}.$$
Clearly, there are other representations for $n = 8$, as well as for values $n > 8$.