A Recursive Formula For Generalized Josephus problem
Let me explain the idea. 
Assume that indices start from 0.
Take N = 6 and K = 3
So initial arrangement looks something like


  0->1->2->3->4->5->0.... like a circle.


After round 1 , '2' is eliminated . 


  0->1->2->3->4->5->0->1...
  
  ....(k)-&gt;0-&gt;1-&gt;2-&gt;3-&gt;
  
  
    here (k) denotes the person is killed
  


Since we ought to start from the last position we killed, so lets look at the updated indices.


  Old Index |  New Index
  
  
    3 | 0
    
    4 | 1
    
    5 | 2
  
  
  and so on.... 
  looking closely we can easily see after each round
  
  
    OldPosition = (newPosition+k)mod N
  


Where N is the number of people left before the round started.

Also OldPosition signifies f(N,K)

and New Position signifies f(N-1,K) as one person has already been killed.

So putting it back to OldPostion = (newPostion+k) mod N
we get

f(N,K) = ( f(N-1,K)+ K ) mod N

But this is done only if u have indices starting from 0. If u want to rearrange for indices starting from 1, you can rearrange it to get the above result.

I found a beautiful paper on this 

http://blue.butler.edu/~phenders/InRoads/MathCounts8.pdf

Hope this helps :)