I am not sure if this was the correct forum...

I need some help with a math question (trying to establish how to work with an inequality division mentioned below)

Someone posted the following problem, and I solved it using the method in the image - but I ended up having to deal with an inequality division and my way around it was very likely not optimal. Can you please have a look and tell me what else can be done? The resultnumericallyis correct (checked it with manual calculations for scaled down versions), but I don't like the inequality division part. I also would wish to ask if the Sum-based recursion is correct (silently I accepted it should be, since it led to the right inequality...)

The problem was stated like this: 100 people are to take slices off a cake. The first takes 1% of what is there (ie of the entire cake), the second takes 2% of what remained, the 3rd takes 3% of what remained etc. Find who will get the largest slice.

Note about my solution: I started by defining the function recursively; any person takes x/100 multiplied by 1-the sum of all the previous f(x), so up to that of x-1.

(Note2: you can try the same for an easier to calculate problem, for example 10 people taking 1/10, 2/10 of what remains etc of the cake. There you end up with x^2+x>=2.7=>xEN:x=2=>x+1=3 and indeed f(3)>f(4)).

Thank you for any help!