You mentioned in a comment that this came up in an algorithmic context. In programming terms, the general expression for this is
reduce(operation, iterable) (of course, this implies that you're dealing with an associative binary operator;
reduce(mean, iterable) isn't going to get the mean of the iterable), e.g.
reduce(or, (f(_) for _ in range(k))). There's also
any(f(_) for _ in range(k)). Those are the Python syntax, but if you're doing psuedo code, this should be clear even to CS people who aren't familiar with Python. In mathematical logic terms, it would be $\exists n:f(n)$.