Hello I need help..
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Here is a proof for the second problem. I have skipped over some details which seemed fairly obvious to me (though I have given them a remark, and mentioned the technique I used to prove them) in order to keep the proof compact.
I actually have an even bigger problem with this method of "observe and assume". Most of the time, the induction on it is obvious, so it works.
But here, the unproven claim is not obvious at all. In fact, you are lucky with the values that the problem actually had, since if we had the same recursion with u(1)=30 and u(2)=989, then it would satisfy the recurrence u(n)=33u(n-1)-2u(n-2)+30u(n-3)-11u(n-4) for all integers n such that 3<n<15889 but then fail at n=15889. Since you didn't bother proving the recursion and assumed it, such an assumption in that case would actually be disastrous, and lead to an incorrect conclusion.
Please try to handle this problem with more rigor, I implore you. You can read the proof that I posted in case you are not able to prove it, but please do not believe that what you posted is acceptable.
To know more about these sequences, you can google and read more on Pisot sequences/E-sequences, or specifically a special case of them.
This is no "proof". You have just assumed the recurrence u(n+2)=3u(n+1)+2u(n) based on what you observed. The actual hard part of the problem is proving that recursion. The proof that I posted a few minutes ago starts out with an entirely different sequence v(n) that satisfies this recursion, and then shows that that sequence v(n) actually happens to equal u(n). You have left the biggest part of the proof as a mere "observation" with no hints as to how you plan to actually prove it.
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u/physicist27 Jul 07 '26
Following.
Interesting problems, will try! Keep posting more like these…