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        riddles >> hard >> THE COEFFICIENT OF x^2
        
(Message started by: pcbouhid on Nov 29^th, 2005, 8:30am)

Title: THE COEFFICIENT OF x^2
Post by pcbouhid on Nov 29^th, 2005, 8:30am

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)

Title: Re: THE COEFFICIENT OF x^2
Post by Joe Fendel on Nov 29^th, 2005, 9:22am

If I had to guess, I'd say [hide](4^(2n-1) - 4^(n-1)) / 3[/hide].

Title: Re: THE COEFFICIENT OF x^2
Post by Eigenray on Nov 29^th, 2005, 10:13am

Well, no reason not to prove your guess:
[hideb]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 a_n=4 for all n>0, so we can rewrite the recurrence as
b' = 4b, c' = 4c + b².
Then clearly b_n = -4ⁿ, so this becomes
c_n+1 = 4c_n + 4²ⁿ,
and c₁=1. This recurrence has the unique solution
c_n = (4^2n-1-4^n-1)/3.[/hideb]

Title: Re: THE COEFFICIENT OF x^2
Post by pcbouhid on Nov 30^th, 2005, 10:31am

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!!!!!!!!!!!! ;D

Title: Re: THE COEFFICIENT OF x^2
Post by srn347 on Sep 17^th, 2007, 6:03pm

What if you replaced n with x?

Title: Re: THE COEFFICIENT OF x^2
Post by ThudanBlunder on Sep 17^th, 2007, 7:14pm

on 09/17/07 at 18:03:02, srn347 wrote:

What if you replaced n with x?

What type of number is n?
What type of number is x?

Title: Re: THE COEFFICIENT OF x^2
Post by srn347 on Sep 17^th, 2007, 8:49pm

Ask pcbouhid.