A patient will have to take a dose of D milligrams of a drug once a...

  

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A patient will have to take a dose of D milligrams of a drug once a day for the rest of their life. It is known from experiments that a fraction, p, of the drug remains in the body after 24 hours. Just after taking the drug on the first day, the patient will have D milligrams in his body, of which pD will be left the next morning. Thus, just after taking the drug on the second day, the patient will have D+pD milligrams in his body. Of this, p(D+pD) will be left the next morning. So, just after taking his medication on the third day, the patient will have D+pD+pD milligrams of the drug in his body. (a) Write a formula for the number of milligrams, UN, of the drug in the patient's body just after he takes the medication on the Nth day. (b) True or False: UN+1 > UN for all N. Explain your answer. (c) Explain why UN approaches a finite number L as N tends to infinity, and find a formula for L in terms of D and p. (d) Why would it be important for the person who decides on the dose D to know this formula? (e) What would happen if p = 1? A patient will have to take a dose of D milligrams of a drug once a day for the rest of their life. It is known from experiments that a fraction, p, of the drug remains in the body after 24 hours. Just after taking the drug on the first day, the patient will have D milligrams in his body, of which pD will be left the next morning. Thus, just after taking the drug on the second day, the patient will have D+pD milligrams in his body. Of this, p(D+pD) will be left the next morning. So, just after taking his medication on the third day, the patient will have D+pD+pD milligrams of the drug in his body. (a) Write a formula for the number of milligrams, UN, of the drug in the patient's body just after he takes the medication on the Nth day. (b) True or False: UN+1 > UN for all N. Explain your answer. (c) Explain why UN approaches a finite number L as N tends to infinity, and find a formula for L in terms of D and p. (d) Why would it be important for the person who decides on the dose D to know this formula? (e) What would happen if p = 1?

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