grandes-ecoles 2022 Q17

grandes-ecoles · France · mines-ponts-maths1__psi Sequences and Series Recurrence Relations and Sequence Properties

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality.
Let $n \in \mathbf{N}$. Show that $P_{n,N}$ is included in $[0,n]^N$ and non-empty for all $N \in \mathbf{N}^*$, that the sequence $(p_{n,N})_{N \geq 1}$ is increasing and that it is constant from rank $\max(n,1)$ onwards.

For $(n,N) \in \mathbf{N} \times \mathbf{N}^*$, we denote by $P_{n,N}$ the set of lists $(a_1, \ldots, a_N) \in \mathbf{N}^N$ such that $\sum_{k=1}^{N} k a_k = n$. If this set is finite, we denote by $p_{n,N}$ its cardinality.

Let $n \in \mathbf{N}$. Show that $P_{n,N}$ is included in $[0,n]^N$ and non-empty for all $N \in \mathbf{N}^*$, that the sequence $(p_{n,N})_{N \geq 1}$ is increasing and that it is constant from rank $\max(n,1)$ onwards.

Paper Questions

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