oh stupid me translating wrong :/
I just meant square number, so let's say you have the problem
The sequence an is defined by ao=a1=1 and an+1=14an - an-1 - 4.
Prove that all terms of this sequence are square numbers.
Is it always that I need a unique way for such problems or are their certain methods that I can use to proof that something is a square number?
and also without a calculat how can I proof that for example 824464 is a square number?
q1: use induction im pretty sure. You COULD use matrix methods, but that would be overly complicated when you go to diagonalise it.
Induction will not work, the reason is that if we started with different that are square numbers, the sequence need not be made up of square numbers.
E.g: gives which isn't square.
Also, matrix methods may not be so useful as is NOT a linear function of and (i.e there is a constant term -4).
I tried calculating the generating function but, if I haven't made a mistake, it doesn't look too nice either.
lol, you HAVE to use a0=1 and a1=1 ... that's your base case lol, then prove for n>2. That's kinda the point of induction... prove the base case.
suppose and are square numbers, then ... (some working)... is a square.
I believe there should be a simple method given that this is some olympiad high school comp.?!?
Your mistake with the matrix method is that the the map is NOT linear. I'm pretty sure that it is NOT true that
...
but like you I believe there should be a simple method given that this is some olympiad high school comp.
Yes I know, but my point is that if you had started with different square numbers for and then the result isn't true (even though the base case is! 4 is a square number).Have you considered "double induction" where there are 2 assumptions, or 2 inductive steps? or have u just gone, nup not even gonna try it coz i have a much better sleeker method.
So we need something better than just a simple induction.
Quote from: kamil9876 on June 05, 2013, 10:54:48 PM
I believe there should be a simple method given that this is some olympiad high school comp.
?!?
Of course this problem is not taken from an actual IMO paper, because I am just preparing for the selection comp, but IMO questions are incredibly hard and not just simple because it is some olympiade high school comp.
Say that again after you solved this question:
CAN YOU READ?
Or did you choose not to read the big red edit I put in, or did you want an ego boost and be a smart ass :D
Have you considered "double induction" where there are 2 assumptions, or 2 inductive steps? or have u just gone, nup not even gonna try it coz i have a much better sleeker method.
I might have made a mistake somewhere, but I cbbs double checking haha.
So why would you want to start with different numbers in this problem..
I'm working on that problem atm, I just got to the point that
ao=a1=1 and an+1=14an - an-1 - 4.
Now let:
bo=b1=1 and bn+1=4bn - bn-1
Now I would need to proof that an=(bn)²
I found this sequence by just plugging in some numbers and realising a sequence in sqrt(a). Now I would just need to proof it, but I don't know how
Geez, can you please stop being so defensive. I'm sorry for reading your post, if you don't want me to read the working and make absolutely no comments about it then delete that part of the post.
As for the double induction - Yes that is why I said "simple induction" won't work, but of course I would like to see if there is some stronger induction that could be done.
Because it shows that straightforward induction won't work (but of course as you mentioned a stronger one may). Please read my other post for an explanation.
Let P(n) be the statement: and
Then Is clearly true since it just says and . Then assuming P(n), you may derive P(n+1) with some straightforward algebra.
I'm still wondering how you guessed the correctly.
And you're such a nice person, labelling it a "guess". I'm sure your opposed to penicillin too, since that was an accident. Btw, if you were actually curious, you can run the numbers too and come to a similar conclusion.Uncalled for, stop bickering and get back to the maths.
I'm working on that problem atm, I just got to the point that
ao=a1=1 and an+1=14an - an-1 - 4.
How did you get to the second equation?
So let P(n) denote the statement:
: and
Thx for the help guys :D, just one question
How did you get to the second equation?
EDIT: Now I really wish I'd made the diagram smaller :(Tip: Add "width=950" in the image tag, so that it should read:
[img width=950]http://i.imgur.com/bZPVR7s.png[/img]