Consider all natural numbers whose decimal expansion has only the even digits $0,2,4,6,8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence then the value of the limit: $\displaystyle \lim_{n\to \infty}\frac{\log a_n}{\log n}=$
(a) $0$.
(b) $\log_5 10$
(c) $\log_210$
(d) $2$
I observe that the sequence $\{a_n\}=\{2,4,6,8,20,22,24,26,28,40,42,44,46,48,60,62,64,68,80,82,84,86,88,200,202,204,206,208,\cdots\}$.
I am not getting the explicit formula for $a_n$, but it shows that the growth is exponential. How to find the exact value of the limit ?
Also I found that $a_n \ge n$ always, from which we get the required limit is $\ge 1$. So option (a) is incorrect.
Any hint. please
Best Answer
Assuming that limit actually exists, take $n=5^k$ then $a_n= 2\times10^{k}$ for all $k\in\mathbb{N}$.
$$\displaystyle \lim_{n\to \infty}\frac{\log a_n}{\log n}=\lim_{5^k\to \infty}\frac{\log (2\times10^{k})}{\log 5^k}=\lim_{5^k\to \infty}{\log_{5^k} (2\times10^{k})}=\log_510$$