**I *think* I got it.**
> **TL;DR**: The answer is $2x$ and $3x-y$.
<sub>Apparently, since I was doing this at a state of "hard-thinking", I wasn't able to put much of this into mind simply because my brain was too exhausted with the problems, so I basically gave up when 2 letters on both sides showed up.</sub>
Now that I cleared my mind a bit more, I noticed something with both variables. $x$ was going up while $y$ was going down.
We can get the differences of each variable:
$$ x - 4x = -3x $$
$$ y - 2y = -3y $$
So now we know both variables have a difference of 3.
How was I able to get the arithmetic means of $10$ and $40$? By doing the same.
$$ 40 - 10 = 30 $$
So if we can get $10$ to $40$ using $30$, then it must be using $6$ ~~intervals (choose a better word for me) ~~ if the means were $5$.
$$ 30 / 6 = 5 $$
So all I had to do then was add $5$ to every mean starting from $10$, which immediately gave the sequence:
$$ 10, 15, 20, 25, 30, 35, 40 $$
So what if we did the same with $x + y$ and $4x - 2y$?
$$ 3 / 3 = 1 $$
$$ x, 2x, 3x, 4x $$
$$ y, 0, -y, -2y $$
$$ x + y, 2x, 3x - y, 4x - 2y $$
So there you have it. The two arithmetic means in this sequence were $2x$ and $3x - y$.
General Sebast1an
·
2022-07-12T11:44:58Z (over 2 years ago)