Question 73
A sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ of natural numbers is defined by the rule
$$
a_{n+1}=a_{n}+b_{n} \quad(n=1,2, \ldots)
$$
where $b_{n}$ is the last digit of $a_{n}$. Prove that such a sequence contains infinitely many powers of 2 if and only if $a_{1}$ is not divisible by 5 .
$$
a_{n+1}=a_{n}+b_{n} \quad(n=1,2, \ldots)
$$
where $b_{n}$ is the last digit of $a_{n}$. Prove that such a sequence contains infinitely many powers of 2 if and only if $a_{1}$ is not divisible by 5 .