Let

be the number of successes in the first 3 trials.

be the number of successes on the possible next 5 trials.

With the assumptions of independence of the trials, we have follows a binomial distribution and follows a truncated geometric distribution.

And you would like to find the distribution of , which is the number of total successes.

The pmf of this is not very nice and I guess you need to calculate for each support point.