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I've known the following theorem.

Theorem: Supposing that $a_{n+1}=\lfloor 1.05\times a_n\rfloor$ for any natural number $n$, there exists $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $20\le a_1\le100$.

Proof: $\{a_n\}$ is a monotonic increase sequence, so let's observe the minimum $n$ such that $a_n\ge100$ for any $a_1$. The observation shows you that you'll always get any one of $100, 101, 102, 103$. Then, you get $$101\to106\to111\to116\to121\to127\to133\to139\to145\to152\to159\to166\to174\to182\to191\to200$$

$$102\to107\to112\to117\to122\to128\to134\to140$$

$$103\to108\to113\to118\to123\to129\to135\to141\to148\to155\to162\to170,$$ so the proof is completed.

Then, here are my questions.

As far as I know, the next question still remains unsolved.

Question1: Supposing that $a_{n+1}=\lfloor 1.05\times a_n\rfloor$ for any natural number $n$, does there exist $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $a_1\ge20\ $?

It is likely that such $N$ would exist, but I'm facing difficulty. I'm also interested in the following generalization.

Question2: Supposing that $\alpha\gt1$ is a real number and that $a_{n+1}=\lfloor \alpha\times a_n\rfloor$ for any natural number $n$, does there exist $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $a_1\ge \frac{1}{\alpha-1} \ $?

Any help would be appreciated.

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  • 1
    $\begingroup$ It may help to observe that $\lfloor1.05 a_n\rfloor=a_n+\lfloor0.05a_n\rfloor$. (Or it may not.) $\endgroup$ – alex.jordan Aug 8 '13 at 15:24
  • $\begingroup$ @alex.jordan: Thank you. This sequence equals an operation to add the quotient, when you divide $a_n$ by 20, to $a_n$. $\endgroup$ – mathlove Aug 8 '13 at 15:37
  • $\begingroup$ How the one's digit changes might be useful, or not. $\endgroup$ – mathlove Aug 9 '13 at 14:55
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If I choose $a_1=23$ then $a_2=24$, $a_3=25$, $a_4=26$, $a_5=27$, $a_6=28$, $a_7=29$, $a_8=30$. Does this answer your question? Sorry for not writing this under comment my reputation is low. In fact I have found that there exit $N$ less than 30 for all $a_1$ between 20 and 100. Here is a python implementation

import math

for seed in range(20,100):
    print('The seed is now: ',seed)
    for i in range(30):
        seed = math.floor(1.05*seed)
        if seed%10 == 0:
            print(i)
            print(seed)
            break

N.B: If $a_1$ is $96$ then $a_2$ is 100.

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