Olympiad Sequence Question

L

Levido

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Dec 22, 2019
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Hi everyone! While there is no section for this type of question I felt it best fit here in advanced algebra. It’s an Olympiad question (the first one so probably a warmup, even more embarrassing :( ) about a series, and I haven’t been able to find a solution anywhere though I’ve been thinking on it for an hour and a half already.

The Question:
4C224C5F-AFB1-42F9-8A20-544B60BA6EEB.jpeg

Here’s an abbreviated working showing my findings and where my confidence I can prove it disappears:
E99B0C86-D99B-4E4A-A32F-4EC77890E1CF.jpeg

Thanks for reading :)
 
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90 minutes on one problem is nothing. Some mathematicians spend their whole carer on one problem and never solve it. Think some more about this problem. Good luck!
 
I definitely will spend more time, I have definitely not given up. I‘ve reached the point where I feel like I’ve used all my problem solving tools though so thought I needed a hint in the right direction, I don’t think I can break the problem down further, not to say I won’t try. For example, although I know un+1 = 2un+1, but I can’t see a way to rearrange a single equation with 3 different terms to one with 2 terms and a constant to prove this. I think there is something that telescopes but I’ve never had to find something like that and prove it before. I will continue to post my findings as I spend more time on it. Thank you for your wishes Jomo :)
 
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I must admit I feel quite stupid at missing this now
B03FAC6A-26FB-44FA-84D4-B33E1EED720A.jpeg
I was typing this up as you replied :)
I’m trying to see where that leads though, I’m trying induction on y now
 
Did you try a proof by induction? I'd do an induction on n.
 
You'll be able to get it, if you just look at the sequence of numbers you have produced. They are very close to certain well-known numbers.
In other words you should be able to see a general rule for the terms of the sequence. You can then prove that general rule for [MATH]x_n[/MATH] using induction.
Then using this formula, it is pretty easy to show that [MATH]y_n[/MATH] is the square of an odd number. You don't need induction for this bit. You can get it just from the expression for [MATH]y_n[/MATH].
(Sorry I didn't see your posts)!
 
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