2. A patient needs to take medicine in pill form, taking a dose of d mg every 12 hours. The medicine has a half life of 6 hours, so every 12 hours the amount of medicine in the patient's body decreases by a factor of In 212 -2 In 2 In(2-2) e = e = e = This also makes sense since 12 hours is two 6 hours half-lives, so the amount decreases by 50% twice every 12 hour period however this exponential/log approach is what we'd need if the time between doses wasn't an integer multiple of the half-life. (a) Let An be the amount of medicine in the patient's body immediately after taking the nth dose. Write out carefully what A1, A2, and A3 are, then write An as a finite geometric series. (b) Use the geometric series partial sum formula to find an explicit, closed-form (that is, no sigma notation or other kind of shorthand) formula for An. (c) Let Bn be the amount of medicine in the patient's body immediately before taking the nth dose. Write out carefully what B1, B2, and B3 are, then write as a finite geometric series. (d) Use the geometric series partial sum formula to find an explicit, closed-form formula for Bn. (e) Suppose we need at least 10 mg of the medicine in the patient's blood for it to be effective. What values of d will guarantee that the medicine remains effective in the long run? (f) Unfortunately, the medicine is toxic if 100 mg or more is ever in their blood. What values of d will guarantee the medicine remains safe in the long run? Hint: For each of (e) and (f), decide which limit lim An or lim B is relevant. n x NX