It is very hard to answer this question because it apparently is based on some sort of multiple choice exercise FOR WHICH WE ARE NOT SUPPLIED THE ALTERNATIVE CHOICES. Moreover, multiple choice problems are inherently questionable because more than one choice may be correct. Nevertheless, we must deal with what we have.
The choice of "afterwards" is grammatical, idiomatic, and plausible.
For the second set of choices, we know two of them, namely "after" and "within." Both are grammatical. In this SPECIFIC case, "after" may not be idiomatic because few native speakers would risk the confusion of the construction "afterwards. After." But in general, "after" would be perfectly good usage. Moreover, I understand the logical objection that in actual fact improvement is likely to occur gradually during a period of study rather than suddenly after the end of that period, but the example concerns "considerable" improvement, which is a binary concept. I suspect that this is a case where the creators of the multiple choice problem created more than one possible correct choice.
With respect to the third set of choices, we cannot comment until the question is edited to specify what choices were given.
EDIT: Upon re-reading, it occurs to me that my original answer presumed too much on the question having been elicited by a multiple choice exercise. The exercise may have just been fill-in-the-blank. In that case, what answers are correct is highly subjective. Except for the potential confusion inherent in the construction of "afterwards. After" the most logical (and perfectly grammatical and idiomatic) choice is " after" because either improvement was considerable or it was not and so occurred at a definite though perhaps unknown instant. If the instructions were to fill in the blank, then "after" is also a suitable choice for the third element of the exercise. Of course, using "afterwards" and following it with "after" twice in close succession is poor style, but this is a site for learners, not budding Walter Paters.
