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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.

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.

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