Distributing a set of integers where set follows a patttern

Question

Forgive incorrect formatting/language. My knowledge of mathematical notation is limited.

Is this possible - and, if so, how would I set about coming up with a solution/solution set given the following logic?

x=a whole number greater than y

y=the number of pieces that I want to divide x into

z=some number less than 1

where the following is true:

    ( y₁ + y₂ +y₃ +y₄ +y₅ ) = x & {

        ( y₁ * z ) = y₂ &
        ( y₂ * z ) = y₃ &
        ( y₃ * z ) = y₄ &
        ( y₄ * z ) = y₅ &

}

and all of y₁...y₅ are whole numbers

extra kudos for a solution so that all y₁...y₅ are easy numbers - that is numbers ending in a 5 or a 0

note: Another alternative solution could involve subtracting rather than taking a percentage of. In other words, something like [10,8,6,4,2]=30, because those numbers add up to 30, and each number is exactly 2 greater than the next. This seems easier to solve, but I still don't know how to write the function for figuring that out.

note: Also, my tags are probably incorrect. Please change if you see fit. Thanks.

Answer 1

It looks like you want $$y_1(1 + z + z^2 + z^3 + z^4) =x$$ which means $$y_1 = x/(1 + z + z^2 + z^3 + z^4)$$ Then you can successively recover $$y_2 = y_1z = xz/(1 + z + z^2 + z^3 + z^4)$$ $$y_3 = y_2z = xz^2/(1 + z + z^2 + z^3 + z^4)$$ $$y_4 = y_3z = xz^3/(1 + z + z^2 + z^3 + z^4)$$ $$y_5 = y_4z = xz^4/(1 + z + z^2 + z^3 + z^4)$$