Question 13
For $\{1,2,3,\ldots,n\}$ and each of its nonempty subsets a unique **alternating sum** is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$ and for $\{5\}$ it is simply $5.$) Find the sum of all such alternating sums for $n=7.$