I have a problem dealing with systems of equations. The equations, in the form of
ax+(a+n)y+(a+2n)z=1+3n
or larger (3+ variables). The coeffecients represent an arithmetic sequence with difference n. So far, I managed to proved that for a 3x3, not matter the coefficients, as long as they follow that pattern:
y=-2x-1
z=x+2

With a 4x4 and a 5x5, I managed to form a generalization. The patterns i pretty simple to see (y,z):

3x3: -2x-1 , x+2
4x4: f-2x-1 , -2f+x+3
5:5: 2g+f-2x-1 , -3g-2f+x+4

The generalization:



I have no idea how to go about proving this, when the number of terms isn't even known. I am thinking that some how, the middle terms all cancel out, so they are irrelevant, or some properties of matrices.

Any help would be great.