in that case in order to passe the tests we shall do to int conversion, will it still be the best practice for this exercice? I used " function summation($n) : int "

That's great. But to better protect against overflow with very big $n, i like to write it as "$n/2 * ($n+1)", this way $n will never multiply with $n+1 before the division.

This is the best practice

Why does this work?

part of your condition for the bounce ($bounce < 0.0) is not valid, it should be $bounce <= 0.0 in your case ;)

//from description condition 2): 0 < bounce < 1 ; When I took out your condition it would look like: 0 <= bounce < 1

But thumbs up for the logaritmic solution.

I like this one, it uses some mathematical formula .