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1983 AIME Problems/Problem 13

Revision as of 21:08, 31 January 2017 by Blue8931 (talk | contribs) (→‎Solution 3)

Problem

For $\{1, 2, 3, \ldots, n\}$ 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 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 $n=7$.

Solution 1

Let $S$ be a non- empty subset of $\{1,2,3,4,5,6\}$.

Then the alternating sum of $S$ plus the alternating sum of $S$ with 7 included is 7. In mathematical terms, $S+ (S\cup 7)=7$. This is true because when we take an alternating sum, each term of $S$ has the opposite sign of each corresponding term of $S\cup 7$.

Because there are $63$ of these pairs, the sum of all possible subsets of our given set is $63*7$. However, we forgot to include the subset that only contains $7$, so our answer is $64\cdot 7=\boxed{448}$.

Solution 2

Consider a given subset $T$ of $S$ that contains 7; then there is a subset $T'$ which contains all the elements of $T$ except for 7, and only those. Since each element of $T'$ has one element fewer preceding it than it does in $T$, their signs are opposite; so the sum of the alternating sums of $T$ and $T'$ is equal to 7. There are $2^6$ subsets containing 7, so our answer is $7 * 2^6 = \boxed{448}$.

Solution 3

Denote the desired total of alternating sums of an $n$ element set with $S_n$. We are looking for $S_7$. Note that all alternating sums of an $n$ element set are also alternating sums of an $n+1$ element set. However, when we go from an $n$ to $n+1$ element set, for each subset with the new element, we are adding the new element and subtracting one of the alternating sums of the $n$ element set. There are $2^n$ subsets of an $n+1$ element set that includes the new element, giving us the following relationship.

$S_{n+1} = S_n + 2^n(n+1) - S_n = 2^n(n+1)$.

For $n = 6$, this becomes $S_ 7 = 2^6(7) = \boxed{448}$.

See Also

1983 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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