Question 1179243
as far as i can tell, the formula will be:


An = .2 * (the sum of .25^(n-x), from x = 1 to x = n)


for example:


when n = 2, the formula becomes:
A2 = .2 * (.25^1 + .25^0) = .25
when n = 3, the formula becomes:
A3 = .2 * (.25^2 + .25^1 + .25^0) = .2625
when n = 4, the formula becomes:
A4 = .2 * (.25^3 + .25^2 + .25^1 + .25^0) = .265625


the dose stabilizes at n = 7 when rounded to 4 decimal places.
the maximum amount of drug in the patient's system is .2667 when n = 7
it remains at .2667 from there on.


the following spreadsheet shows you the progression.


per the formula, when n = 7, the amount of drug in the patient's system will be:


A7 = .2 * sum(.25^(9-x) from x = 1 to x = 7) which becomes:


A7 = .2 * (.25^6 + .25^5 + .25^4 + .25^3 + .25^2 + .25^1 + .25^0) which becomes:


A7 = .2666503906


round to 4 decimal places and you get A7 = .26667.


the following spreadsheet does the calculations for each n.


for example:


when n = 1, h = 0 and A1 = .2 = .2000
when n = 2, h = 4 and A2 = .25 * A1 + .2 = .25 * .2 + .2 = .25 = .2500
when n = 3, h = 8 and A3 =  .25 * A2 + .2 = .25 * .25 + .2 = .2625 = .2625
when n = 4, h = 12 and A4 = .25 * A3 + .2 = .25 * .2625 + .2 = .265625 = .2656
when n = 5, h = 16 and A5 = .25 * A4 + .2 = .25 * .265625 + .2 = .26640625 = .2664
when n = 6, h = 20 and A6 = .25 * A5 + .2 = .25 * .26640625 + .2 = .2666015625 = .2666
when n = 7, h = 24 and A7 = .25 * A6 + .2 = .25 * .2666015625 + .2 = .2666503906 = .2667


as you can see from the spreadsheet, the value remains at .2667 from there on when rounded to 4 decimal places.


here's the spreadsheet calculation for n = 1 to 112.


you can see that the amount of drugs in the patient's system stabilizes at n = 7 when the maximum amount in the patient's system is .2667 and remains at .2667 from there on.


<img src = "http://theo.x10hosting.com/2021/042501.jpg">


you were asked what will be the concentration of drug in the patient's blood immediately after the nth dose.


that will be .2 * (the sum of .25^(n-x), from x = 1 to x = n), as best i can determine.