Background: A dosage of a certain drug is administered to an individual. Assume

Question

Background: A dosage of a certain drug is administered to an individual. Assume that the drug is administered intravenously, so that the concentration of the drug in the bloodstream jumps almost immediately to its highest level. The concentration of the drug then decays exponentially.

Specifics: A doctor prescribes a 240 milligram (mg), pain-reducing drug to a patient who has chronic pain. The medical instruction reads that this drug should be taken every 4 hours. After 4 hours, 60% of the original dose leaves the body.

1. Show that the amount of medicine in the patient's bloodstream after Nth dose can be expressed by a Geometric Series. Use sigma notation to express the series.

Explanation / Answer

the process can be modeled by equation

concentarion of drug at time t is given by C(t) = A.exp(-kt)

after 4 hours 60% of dose leaves the system, so 40% remains.

initial amt. given is 250mg so after 4 hrs, the amount left will be 0.4(240) = 96mg

at t = 0, C = 240 ----> A = 240

at t = 4, C = 96 ----> k=0.2291 (as already calculated A = 240)

so the equation becomes C(t) = 240exp(-0.2291t) ...(C in mg and t in hrs)

now consider f(x) = ex

the maclaurin series of this function will be given by

f(x) = f(0) + f'(0).x + (1/2!).f''(0).x2 + (1/3!).f'''(0).x3 + ......

now for f(x) = ex , f(0) = f'(0) = f''(0) = f'''(0) are all equal to 1

put this in equation, you will get

f(x) = ex 1 + x + (x2/2!) + (x3/3!) + ....

or, f(x) (xn/n!) ... (n from 0 to )

which is an infinite geometric series

so put -0.2291t in place of x in above equation you will see that the the conc of drugs after Nth dose will be expressed as an sum of geometric series the answer being

C(t) = 240(xn/n!) .... (n going fom 0 to N)

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