ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frogstarts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and stays there. Let an be the number of distinct paths of exactly n jumps ending in E. Then, what is the value of a2n-1?

• Firstly, notice that the shortest path from A to E involves exactly 4 jumps.
  4 is even number. We can prove that the frog always makes even number of
  jumps to reach vertex E.
  Assume we have some path from A to E. If at some vertex our path turns back
  then later it must turn back again, so this vertex is visited twice.
  Such way the evenness of length of the path can't be corrupted. Thus, any
  possible path has even length.
  Since (2n-1) is odd, there is no path from A to E that frog can follow for (2n-1) jumps.
  Thus, a(2n-1) = 0
  ANSWER
  A(2n-1) = 0
• 5 years agoHelpfull: Yes(1) No(0)