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Chapter 3: Problem 33
\((x, y) \in R\) if \(|x-y|=2\)
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Rewrite the sum $$ \sum_{k=1}^{n} C_{k-1} C_{n-k} $$ replacing the index \(k\) by \(i,\) where \(k=i+1\).
\( x_{1}, x_{2}, \ldots, x_{n}, n \geq 2,\) are real numbers satisfying
\(x_{1}
Prove that if \(n\) is an odd integer, $$ \left.\mid \frac{n^{2}}{4}\right\rfloor=\left(\frac{n-1}{2}\right)\left(\frac{n+1}{2}\right) $$
For the sequence a defined by \(a_{n}=\frac{n-1}{n^{2}(n-2)^{2}}, \quad n \geq 3\) and the sequence \(z\) defined by \(z_{n}=\sum_{i=3}^{n} a_{i}\). Is \(z\) nondecreasing?
Let \(X\) be the set of positive integers that are not perfect squares. (A
perfect square \(m\) is an integer of the form \(m=i^{2}\) where \(i\) is an
integer.) Concern the sequence s from \(X\) to \(\mathbf{Z}\) defined as follows.
If \(n \in X,\) let \(s_{n}\) be the least integer \(a_{k}\) for which there exist
integers \(a_{1}, \ldots, a_{k}\) with \(n
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