1983 AIME Problems/Problem 13
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Problem
For 
and each of its non-empty subsets, an alternating sum is defined as follows. Arrange the number in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for 
is 
and for 
it is simply 
. Find the sum of all such alternating sums for 
.
Solution
Let 
be a non- empty subset of 
.
Then the alternating sum of plus the alternating sum of with 7 included is 7. In mathematical terms, 
. This is true because when we take an alternating sum, each term of has the opposite sign of each corresponding term of 
.
Because there are 
of these pairs, the sum of all possible subsets of our given set is 
. However, we forgot to include the subset that only contains 
, so our answer is 
.
See also
| 1983 AIME (Problems • Resources) | ||
| Preceded by Problem 12 | Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
