Homework SourseBackground A dosage of a certain drug is administered to an
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.
Solution
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)
