Recently in a contest a question was asked as under.

We define a lead element of a set $\{a_1,a_2,a_3, \cdots a_n\}$ as $$l(\{a_1,a_2,a_3, \cdots a_n\})=|a_1-a_2+a_3-a_4 \cdots(-1)^{n-1}a_n|$$

Now let $X_n$ be the set of natural numbers from $1$ to $n$.

Then find the sum of all the lead elements of all the non empty subsets of $X_n$ in terms of $n$?

I guess some recursion relation will build up ,but I am not able to get the solution.

  • 2
    $\begingroup$ Have you tried writing out the first few terms, finding a pattern, then trying to prove it by induction? $\endgroup$ – rogerl Nov 1 '14 at 13:47
  • 1
    $\begingroup$ what do you use for the ordering of elements needed to define the lead elements of a subset? $\endgroup$ – David Holden Nov 1 '14 at 13:52
  • $\begingroup$ The same order of 1 to n $\endgroup$ – Love Everything Nov 1 '14 at 13:54
  • $\begingroup$ So its not a set, its an ordered sequence is it? If so, you could group the difference of adjacent elements together and see what happens for even and odd $n$ separately $\endgroup$ – Macavity Nov 1 '14 at 14:20
  • $\begingroup$ @Macavity kinda ..but the question used the word set $\endgroup$ – Love Everything Nov 1 '14 at 14:22

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