An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.
joybangla wrote:
An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.
Suppose on the contrary that such a sequence exists with all its numbers odd.
Looking at the sequence $y_n=\left\lfloor\frac{x_n}{10}\right\rfloor$, it's easy to see that $x_n$ is increasing and that $y_{n+1}\in\{y_n,y_n+1\}$
So $\{y_n\}_{n\ge 1}=[y_1,+\infty)$ and so $\exists p$ such that $y_p$ contains only odd nonzero digits. And so $x_{p+1}$ can no longer have same parity than $x_p$, hence the claim.