What is the mathematical formula?

Question

Mathematics

Greetings,
I am developing a lottery and I am faced with the fact that I cannot find a formula for dividing the winnings between the participants, I hope for your help, here's the essence: the
2nd one gets the winnings 2 times less than the first
3rd one gets 2 times less than the 2nd
Winning: 100
winnings places: 3
Should be like this x + 2x + 4x = 100
If there are 4 winning places:
x + 2x + 4x + 8x = 100
If there are 5 winning places:
x + 2x + 4x + 8x + 16x = 100
If there are 6 winning places:
x + 2x + 4x + 8x + 16x + 32x = 100
I can't figure out what formula can be used to automatically calculate if the number of winning places varies, I hope for help from the community!

Answer 1

If n is the number of participants
1x+2x+4x+...+2 ^n-1 x = (2 ⁿ -1)x = 100
x = 100/(2 ⁿ -1)
Each winner gets ( i - taken place)
P _i \u003d 2 ^n-i x \u003d 100 * 2 ^n-i / (2 ⁿ -1)

Answer 2

x = s / (2^(n-1) - 1), where s is the winning amount; n is the number of "seats".

Answer 3

And you try to go from the opposite. For example - the first group of winners receives "half of the prize fund". The next group is half of the remaining half, and so on. The problem will only be to set the point after which we stop dividing the prize.

Answer 4

2^n-1, where n is the number of winning places

Answer 5

In a loop from 0 to the number of places-1, you add 2 to the degree of iteration of the loop. For example for 6 places

Sting S = "";
for(int i=0;i<6;i++) {
   S+=(int)(Math.pow(2,i))+"x";
}

Answer 6

1 + 2 + 4 + ... = 2⁰ + 2¹ + 2² + ... = 2ⁿ - 1
(2ⁿ - 1)x = 100
x = 100 / (2ⁿ -
1
) participants x = 100 / (2⁴ - 1) = 100 / 15
Winnings: 100/15, 2*100/15, 4*100/15, 8*100/15