Question 54
Let $(a_n)_{n \in \mathbb{N}^*}$ be a sequence such that
$$
2(a_1 + a_2 + \dots + a_n) = n a_{n+1}, \quad \text{for all } n \ge 1.
$$
a) Prove that $(a_n)_{n \in \mathbb{N}^*}$ is an arithmetic sequence.
b) If $[a_1] + [a_2] + \dots + [a_n] = [a_1 + a_2 + \dots + a_n]$, for all $n \in \mathbb{N}^*$, prove that the terms of the sequence are all integers (here, $[x]$ denotes the floor of $x$).
$$
2(a_1 + a_2 + \dots + a_n) = n a_{n+1}, \quad \text{for all } n \ge 1.
$$
a) Prove that $(a_n)_{n \in \mathbb{N}^*}$ is an arithmetic sequence.
b) If $[a_1] + [a_2] + \dots + [a_n] = [a_1 + a_2 + \dots + a_n]$, for all $n \in \mathbb{N}^*$, prove that the terms of the sequence are all integers (here, $[x]$ denotes the floor of $x$).