How to find the minimum number of states required by a DFA Given a language $L = \left\{ a^{nk+1} | n &gt; 0 \right\}$ and $k$ is an integer constant. 

How to show that a DFA for this language must have $k+2$ states or more states using minimum state lemma.

By minimum state lemma I mean the number of minimum state a DFA has is the number of pairewise distinguishable states. I have constructed a set of pairewise distinguishable string ${a, aa, aaa, ... a^{k+2}}$ with respect to L and found that I can not add anymore strings to it. But I don't know how to prove this string has the maximum number of pairewise distinguishable strings.
You can prove that your set is maximal in (at least) two different ways:


Show that every string is equivalent to one of the strings in your set.
Construct a DFA for the language having the same number of states as are strings in your set.


In your case both approaches are not too difficult.

Note, however, that the question only asks you to show that every DFA for the language must contain at least $k+2$ states. For this there is no need to show that your collection is maximal. If you find $k+2$ pairwise inequivalent strings, then it follows that every DFA for the language must contain at least $k+2$ states. If the collection is not maximal, all it means is that your bound isn't tight.