Series question

Question

A certain antibiotic is administered orally to a patient. As the liver filters the drug out of the patient’s bloodstream, only 20% remains after eight hours.

(i) If the patient takes a 100mg dose each eight hours, how much of the drug remains in the patient’s bloodstream immediately after the second and third doses?

(ii) Use a limit as n approaches ∞ to approximate how much of the drug remains in the patient’s bloodstream immediately after the nth dose, for n large.

Answer 1

The sequence for which you are looking is: 100, 120, 124, 1 +24.8, 124.96...

It is given by the difference equation: a_n+1=.2a_n+100

The solution of this equation is a_n=.2ⁿ + 100/.8 which has limit 125.

The above answer is not quite correct!

The given sequence is correct and the difference equation is correct.

The correct general solution is 100*.2ⁿ+100*(1-.2ⁿ)/(1-.2) for n≥0

The limit as n->∞ is 125.

Sorry for the initial error.