An intuitive and telling approach to this is to find a functional identity (see note at the end) that the random number $X$ of downloads necessary to get an uncorrupted file satisfies. The everyday situation you describe amounts to the following:
- With probability $p=0.2$, $X=1$ (first file uncorrupted).
- With probability $1-p=0.8$, $X=1+Y$, where $Y$ is distributed like $X$ (first file corrupted, then continue with the next files).
Thus, $E[Y]=E[X]$ hence
$$E[X]=p\cdot1+(1-p)\cdot(1+E[X]),
$$
from which the arch-classical formula $E[X]=1/p$ follows.
Note that this also yields the full distribution of $X$, for example, for every $|s|\leqslant1$, $g(s)=E[s^X]$ is such that $E[s^{Y}]=g(s)$ hence $g(s)$ must solve the corresponding identity $$g(s)=p\cdot s+(1-p)\cdot s\cdot g(s),
$$
hence
$$
\sum_{n\geqslant0}P[X=n]s^n=g(s)=\frac{ps}{1-(1-p)s}=ps\sum_{n\geqslant0}(1-p)^ns^n,
$$
from which $P[X=n]=p(1-p)^{n-1}$ follows, for every $n\geqslant1$.
Note: Since some user was kind enough to upvote this a long time after it was written, I just reread the whole page. Frankly, I found appalling the insistence of a character to confuse binomial distributions with geometric distributions, but I also realized that the functional identity referred to in the first sentence of the present answer had not been made explicit, so here it is.
The distribution of the number $X$ of downloads to get an uncorrupted file is the only solution of the identity in distribution $$X\stackrel{(d)}{=}1+BX,$$ where the random variable $B$ on the RHS is independent of $X$ on the RHS and Bernoulli distributed with $$P(B=0)=p,\qquad P(B=1)=1-p.$$
This merely summarizes the description in words at the beginning of this post, and allows to deduce all the mathematical results above. This also yields a representation of $X$ as
$$X\stackrel{(d)}{=}1+\sum_{n=1}^\infty\prod_{k=1}^nB_k,\qquad\text{with $(B_k)$ i.i.d. and distributed as }B.$$
Finally, note that every positive integer valued random variable $X$ can be represented as the sum of such a series for some independent sequence of Bernoulli random variables $(B_k)$, but that the distribution of $B_k$ being independent on $k$ characterizes the fact that the distribution of the sum $X$ is geometric.