Why arithmetic progression?

I've just found a generalization of this problem, but I doubt there'll be a rigorous solution for this one.

Let $k \ge 3$ be an integer, $a_1>a_2>...>a_k$ be fixed positive integers in an arithmetic progression, $x_1,x_2,...,x_k$ be non-negative reals so that $x_1+x_2+...+x_k=m$ and $a_1x_1+a_2x_2+...+a_kx_k=n$ . Find the minimum and maximum values of $a_1^2x_1+a_2^2x_2+...+a_k^2x_k$ .

The problem here is that those $a$ must be in an arithmetic progression, and that makes me feel awkward. In the case of the maximum, that does not matter. In the case of the minimum though, without that I'd be killed by a lot of functions that goes nowhere!

So is the "arithmetic" thing necessary for a rigorous solution, and can anyone give me a solution of this extension? I'm pretty sure I burnt lots of my brain cells for a month already just because of this.

Any ideas will be appreciated!

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$