Also, for this kind of question you frequently either get an "exact" answer which is intractable (the formula is just too hairy), or some approximations and/or bounds which are in much simpler form. So, what are you looking for?

Did you see this paper jstor.org/stable/1427566 ?

@user1946334 - then what you want to prove is probably wrong, because the "typical" longest sequence of 0's should be of length roughly $\log_2 n$. Anyhow, see the links below.

Do you mean that the entries of your sequence are independent, and the 0's and 1's are equiprobable (i.e., 0 w/p 1/2, 1 w/p 1/2)?

You are mostly interested in the case $r\to0$? Or is $r$ just fixed?

Yes, that's correct.

@SebastianGoette: he asked "within time $t$", not "at time $t$"