Cities and Induction. Trying to find a dead end. There are  ($n$ > 1) cities and every pair of cities is connected by exactly one road. The road can go only from A to B, only from B to A, or in both directions. 
The goal is to find a dead-end city, if it exists, i.e., a city x to which there is a direct one-way road from every other city, but there is no direct road going from X to any other city. You are allowed to make only one type of question – “Is there a direct road going from city A to city B?” The answer to this question will be a “Yes” or a “No”. Use mathematical induction to show that if there are  cities then one can find a dead-end city, if there is one, using at most 2(−1) questions.
My thoughts:
Base C:  = 2
2(2−1)=2 questions. Ask if there is a road from A to B. And ask if there is a road from B to A.
Induction H: Assume that the claim is true when  = , for some k > 1.
2(−1)
IS: We want to prove that the claim is true when n = k + 1
I also noted that there can only be a maximum of one dead-end city. But I'm unsure of where to proceed. 

Lemma: Given this city structure with $n$ cities, you can determine exactly one candidate to be the dead end city with $n-1$ questions.
Pf: By induction.  Easily done with $2$ cities and one question.  For $k+1$ cities, choose two called $A$ and $B$.  Ask if there is a road from $A$ to $B$.  If there is, $A$ cannot be the dead-end city, otherwise $B$ cannot be the dead-end city.  By the induction hypothesis, you can whittle the remaining $k$ candidates down to 1 with another $k-1$ questions.
Proof of proposition: Given $n$ cities, we can determine a candidate city $X$ to be the dead end city in $n-1$ questions.  Ask whether there is a road from $X$ to each of the remaining $n-1$ cities.  If none of them are, then $X$ is a dead end, otherwise there is no dead end city.  This requires a maximum of $2(n-1)$ questions.
