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wu :: forums    riddles    hard (Moderators: ThudnBlunder, towr, william wu, Icarus, Eigenray, Grimbal, SMQ)    THE COEFFICIENT OF x^2 « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: THE COEFFICIENT OF x^2  (Read 1235 times) pcbouhid Uberpuzzler     Gender: Posts: 647 THE COEFFICIENT OF x^2   « on: Nov 29th, 2005, 8:30am » Quote Modify Find the coefficient of x^2 upon the expansion and collecting of the terms in this expression:   ((((x - 2)^2 - 2)^2 - 2)^2 - .....- 2)^2   ------------------------------------                    (n times) « Last Edit: Nov 29th, 2005, 8:34am by pcbouhid » IP Logged Don´t follow me, I´m lost too. Joe Fendel Junior Member     Posts: 68 Re: THE COEFFICIENT OF x^2   « Reply #1 on: Nov 29th, 2005, 9:22am » Quote Modify If I had to guess, I'd say (4(2n-1) - 4(n-1)) / 3. IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: THE COEFFICIENT OF x^2   « Reply #2 on: Nov 29th, 2005, 10:13am » Quote Modify Well, no reason not to prove your guess: hidden: If a'+b'x+c'x2 = (a-2+bx+cx2)2 (mod x3), then a'=(a-2)2, b'=2(a-2)b, and c'=2(a-2)c+b2. If we start with (a0,b0,c0) = (0,1,0), then (a1,b1,c1) = (4, -4, 1). Since a1=4, we have an=4 for all n>0, so we can rewrite the recurrence as b' = 4b,  c' = 4c + b2. Then clearly bn = -4n, so this becomes cn+1 = 4cn + 42n, and c1=1.  This recurrence has the unique solution cn =  (42n-1-4n-1)/3. IP Logged pcbouhid Uberpuzzler     Gender: Posts: 647 Re: THE COEFFICIENT OF x^2   « Reply #3 on: Nov 30th, 2005, 10:31am » Quote Modify It seems a good team work with two members: the first make a guess, and the second, prove the guess. Next time, in separate rooms!!!!!!!!!!!!   IP Logged Don´t follow me, I´m lost too. srn437 Newbie the dark lord rises again....     Posts: 1 Re: THE COEFFICIENT OF x^2   « Reply #4 on: Sep 17th, 2007, 6:03pm » Quote Modify What if you replaced n with x? IP Logged ThudnBlunder wu::riddles Moderator Uberpuzzler The dewdrop slides into the shining Sea     Gender: Posts: 4489 Re: THE COEFFICIENT OF x^2   « Reply #5 on: Sep 17th, 2007, 7:14pm » Quote Modify on Sep 17th, 2007, 6:03pm, srn347 wrote: What if you replaced n with x? What type of number is n? What type of number is x?     IP Logged THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you. srn437 Newbie the dark lord rises again....     Posts: 1 Re: THE COEFFICIENT OF x^2   « Reply #6 on: Sep 17th, 2007, 8:49pm » Quote Modify Ask pcbouhid. IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium => hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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