Math Question, not sure if it has a solution
Hi I have a question I will try to explain it as best as I possibly can.
Ok I have sum that look like this: (a*x^2 + b*y^2 + c*z^2) a, b, c are constants and are positive rational numbers x, y, and z are natural numbers I am trying to find a method to choose values for x, y, and z so I can find all the sums in increasing order. a, b, and c are fixed constants. So for example if a = 1, b = 1/4, c = 1/9, then I have: (1*x^2 + (1/4)*y^2 + (1/9)*z^2) 1st Sum: 1.36, with x = 1, y = 1, z = 1 2nd Sum: 1.694 with x = 1, y = 1, z = 2 3rd Sum: 2.11 with x = 1, y = 2, z = 1 4th Sum: 2.25 with x = 1, y = 1, z = 3 5th Sum: 2.44 with x = 1, y = 2, z = 2 6th Sum: 3 with x = 1, y = 2, z = 3 7th Sum: 3.028 with x = 1, y = 1, z = 4 8th Sum: 3.36 with x = 1, y = 3, z = 1 9th Sum: 3.472 with x = 1, y = 3, z = 2 10th Sum: 3.778 with x = 1, y = 2, z = 4 Of course I can different values for a, b, and c. Let's say a = 1/4, b = 2, c = 1/16 so then I'd have: ((1/4) * x^2 + 2*y^2 + (1/16)*c^2), and I'd have to find values for x, y, and z so that the sums would be in increasing order. Is it possible? And if so any ideas, suggestions? I figure maybe someone here may be familiar with these types of problems. This isn't from a math book or from a math course so I'm not sure where to start. It came up in something else I was doing and I figured it would be fun to find a solution, I just don't want to run around in circles forever if it isn't possible heh. I don't really have a starting point. I tried to find a pattern but I didn't see one. I hope I didn't make any mistakes, it's really late. Thanks. Edit: If I put this in the wrong section feel free to move it. I am not sure where this belongs so I just posted in general. 
hmm..
well since a,b and c are all nonnegative and x,y,z are also nonnegative Therefore set y,z to zero and keep incrementing x. The sum will always be increasing. (Infact fix any two of (x,y,z) to any nonnegative value and keep on incrementing the third variable, as long as the coefficient of this third variable isnt zero).  AI P.S > Am i missing something? 
Quote:

output, then sort.
a=1 : b=1/4 : c=1/9
for z=1 to Zmax step 1 for y=1 to Ymax step 1 for x=1 to Xmax step 1 Sum = (a*x*x + b*y*y + c*z*z) append table, Sum, x, y, z next x next x next x Sort table, Sum 
a=1 : b=1/4 : c=1/9
for z=1 to Zmax step 1 for y=1 to Ymax step 1 for x=1 to Xmax step 1 Sum = (a*x*x + b*y*y + c*z*z) append table, Sum, x, y, z next x next y next z Sort table, Sum 
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