Find an exact formula for this problem

[John112]

You have unlimited supply of marbles, each marble with it's own number(#1, #2, #3, etc...).

At 11pm you add 100 marbles to a empty bucket ( marbles #1 through #100) and then remove marble #1 from the bucket immediately ( so you're adding 99 marbles).

At 11:30pm you add another 100 marbles (marbles #101 through #200) and then remove marble #2 from the bucket.

At 11:45pm you add another 100 marbles (#201 through #300) and remove marble #3.

The process continues, at each step you divide the remaining time in half, add the next 100 marbles and remove exactly one marble.


Find an exact formula for s(t): the number of marbles in the bucket as a function of t, where t is the time in minutes before midnight. (notice s(t) is a step function, for example its value doesn't change for 60≤ t<30)



I tried to a define a single function that would produce solution for this, but It only produces certain terms. Then I tried to define s(t) as a step function, but the problem with that is as the remaining time gets divided in half each time, you can have infinitely many intervals for the step function. Since you can have less than 1 minute. Do can I overcome this? Or should I just try a different approach?

 

[voko]

You could define a sequence of times Tn: at n = 1, time T1 = 60, at n = 2, time T2 is 30, etc.

Then you could define Sn: at n = 1, S1 = 99, etc.

Then you have to figure out how to define s(t) in terms of Tn and Sn :)

 

[HallsofIvy]

At time 1 (11:00), you have 100- 1= 99 marbles.
At time 2 (11:30), you have 200- 2= 198 marbles
At time 3 (11:45), you have 300- 3= 297 marbles

At time n (12:00- 1/2^(n+1) minutes) you have 100n- n= 99n marbles.

 

[LCKurtz]

At time 1 (11:00), you have 100- 1= 99 marbles.
At time 2 (11:30), you have 200- 2= 198 marbles
At time 3 (11:45), you have 300- 3= 297 marbles

At time n (12:00- 1/2^(n+1) minutes) you have 100n- n= 99n marbles.

Don't you mean$$
12:00-\frac {60}{2^{n-1}}$$?