Plus and minus

Task number: 3325

Let \(S_n\) denote the set of all integers that can be written in the form \(\pm1\pm2\pm3\ldots\pm n\) (This is the sum of \(n\) numbers, where we replace each \(\pm\) either with the sign \(+\) or \(-\) independently of the others). Prove the following assertions:

  • Variant

    For all \(x\in S_n\): \(\displaystyle -\frac{n(n+1)}2\le x\le \frac{n(n+1)}2\).

  • Variant

    All numbers in \(S_n\) have the same parity (either all are even, or all are odd). How does this parity depend on \(n\)?

  • Variant

    All integers fulfilling the preceding two conditions lie in the subset \(S_n\).

Difficulty level: Moderate task
Proving or derivation task
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