how do i create a geometric sequence if someone is taking 500mg of a drug every 8 hours and has to do so for 10 days. every 8 hours, their kidneys eliminate 65% of this drug. ill be completing a table for the amount of drug in the body just after each 500mg does has been given

Question

anonymous · Answer

if it helps, the equation that shows the inital dose remaining:
$$A = 500e ^{-0.131t}$$

anonymous · Answer

After first dose:$$A_1=500$$
After second those:$$A_2=A_1·(1-0.65)+500=500·(1-0.65)+500$$
After third dose:$$A_3=A_2·(1-0.65)+500=500·(1-0.65)^2+500+500·(1-0.65)+500$$
In general, after the n-th intake:$$A_n=500·[0.35^{n-1}+0.35^{n-2}+... +0.35^1+0.35^0]$$This can be expressed as:
$$A_n=500\sum_{k=0}^{n-1}0.35^k$$
and 0.35^k is a geometric sequence whose sum is:
$$S_n=\sum_{k=0}^{n-1}0.35^k=\frac{ 1-0.35^n}{ 1-0.35}=\frac{ 1-0.35^n }{ 0.65}$$Then:$$A_n=500·S_n=500·\frac{ 1-0.35^n }{ 0.65 }$$

anonymous · Answer

After third dose, it should say:
$$A_3=A_2⋅(1−0.65)+500=500⋅(1−0.65)^2+500⋅(1−0.65)+500$$

anonymous · Answer

thank you @CarlosGP

anonymous · Answer

you are welcome