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Registered Member #2140
Joined: Tue May 26 2009, 09:16PM
Location:
Posts: 53
I know it isn't exactly electronic-y, but figured I would ask here, where there's a lot of smart people.
I have an assignment, and it is finding patterns within linear equations. The linear equation's coeffecience have a common difference. e.g:
x+2y=3 (difference of 1) 2x-y=-4 (difference of 2)
The pattern here is for any 2x2 equation, with a common difference, the solution is always (-1,2). For a 3x3, it is y=-2x-1, z=x+2.
The problem I have is extending it. This is what I noticed, for the y values:
3x3: -2x-1 4x4: u-2x-2 5x5: 2f+u-2x-3
Where the variable progression is x,y,z,u,f.
The pattern is that everytime, the last (n-3) variable - 1 is added. E.g, -2x-1 is a 3x3. To go to 4x4, the coefficient would be 1 (4-3), so I add u-1. 4x4 to 5x5 is a coefficient of 2 (5-3), so I add 2f-1.
Because this is so arbitrary, with out even a concrete number of terms, how would you go about proving it? My guess is somehow, a bunch of middle terms cancel out, so you only have to worry about the first and last equation.
Registered Member #2919
Joined: Fri Jun 11 2010, 06:30PM
Location: Cambridge, MA
Posts: 652
What do you mean "coefficients have a common difference?" Do you mean that if we write the system as a matrix {a_ij}, the rows of the matrix from arithmetic sequences?
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Math Proof
