Let $a_n = 1 . . . 1 $ with $3^n$ digits. Prove that $a_n$ is divisible by $3a_{n−1}$.

Question

Let $a_n = 1 . . . 1 $ with $3^n$ digits. Prove that $a_n$ is divisible by $3a_{n−1}$. Is there any way to solve this question without mathematical induction?

Answer 1

$a_n=\dfrac{10^{3^n}-1}9$,

so $\dfrac{a_n}{a_{n-1}}=\dfrac{10^{3^n}-1}{10^{3^{n-1}}-1}=10^{2\cdot3^{n-1}}+10^{3^{n-1}}+1,$

which is $\equiv1+1+1\equiv0\pmod3.$