Proizvolov's identity
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The following defines Proizvolov's identity.
Take the first 2N natural numbers:
- 1, 2, 3, ..., 2N − 1, 2N.
Choose any subset of N numbers and arrange them in an increasing sequence:
Arrange the remaining numbers in a decreasing sequence:
Then the sum
is always equal to N2.
[edit] Example
Take for example N = 3. The set of numbers is then {1,2,3,4,5,6}. Select three numbers of this set, say 2, 3 and 5. Then the sequences A and B are:
- A1 = 2, A2 = 3, and A3 = 5;
- B1 = 6, B2 = 4, and B3 = 1.
The sum is
- | A1 − B1 | + | A2 − B2 | + | A3 − B3 | = | 2 − 6 | + | 3 − 4 | + | 5 − 1 | = 4 + 1 + 4 = 9,
which indeed equals 32.
[edit] External link
- Proizvolov's Identity at cut-the-knot.org
