Question 139
Let $a$ be a given natural number. We consider the sequence $(x_n)_{n \ge 1}$ defined by $x_n = \frac{1}{1+na}$, for every natural nonzero number $n$.
Prove that, for every natural number $k \ge 3$, there are non-zero natural numbers $n_1 \lt n_2 \lt \dots \lt n_k$ so that the numbers $x_{n_1}, x_{n_2}, \dots, x_{n_k}$ are the consecutive terms of an arithmetic progression.
Prove that, for every natural number $k \ge 3$, there are non-zero natural numbers $n_1 \lt n_2 \lt \dots \lt n_k$ so that the numbers $x_{n_1}, x_{n_2}, \dots, x_{n_k}$ are the consecutive terms of an arithmetic progression.