What is the formula for the progressive coefficient?

Question

Mathematics

Given:
There are 4 numbers, the sum of which is 200 + 300 + 50 + 50 = 600.
You need to multiply each number by its own coefficient so that instead of 600, the sum of all these four numbers becomes 800.

Yes, this is elementarily solved:
200 x 1 = 200
300 x 0.66666 = 200
50 x 4 = 200
50 x 4 = 200
total: 800

Question: tell me the formula so that the strength of the coefficient is different, let's call it "progressive"?
In other words, the more the coefficient is different from unity, the greater the effect it has on the number multiplied by it.

For example,
200 x 1 = 200
300 x 0.9 = 200
50 x 1.3 = 200
50 x 1.3 = 200

Don't ask where I got the coefficients 0.9 and 1.3. The coefficients 0.9 and 1.3 are just taken to give an idea of ​​what I need.

Answer 1

Something with the exponent?
200 * e ^{(k ₀ - 1)} + 300 * e ^{(k ₁ - 1)} + 50 * e ^{(k ₂ - 1)} + 50 * e ^{(k ₃ - 1)}
When k == 1, the power is obtained 0, and the factor e ⁰ is 1.
To increase steeply with a slight increase, you can add a factor:
x * e ^{2k - 2}

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