Question 1144121: Help me please. I do not know where to start.
1. Your doctor prescribes a daily 20 mg dose of a popular allergy medication, No-Drip. Roughly 24 hours after a first dose of the medicine is used, 40% of the No-Drip has been flushed from the body. You take 20 mg of No-Drip once a day. Make a table to show the maximum amount of No-Drip in your system each day for six days. Express each day's amount as a sum. Then, sketch a graph of the maximum No-Drip and the day number for at least fourteen days.
2. Find the partial sum of the first seven terms in the series described in question 1.
3. What do you notice about the graph of the amount of No-Drip in your system over fourteen days? What does this mean in the context of the problem? Confirm your claim analytically.
4. Make a conjecture about the convergence of arithmetic and geometric series.
5. Explore the sequence of partial sums for these three series using tables and graphs. In each case, will the series converge or diverge?
6. Write a formula for the sum of a convergent geometric series with first term t1 and common ratio r, where |r| < 1.
Answer by ikleyn(52790)   (Show Source):
You can put this solution on YOUR website! .
As the given problem instructs you, solving it is the same as to go forward with closed eyes.
I will solve it by MY WAY : I will go with OPENED EYES.
Solution
1-st day, 8:00 am in the morning. You take 20 mg of the medication.
Next day, (day 2), 7:59 am. Of these 20 mg, only 1-0.4 = 0.6 of 20 mg, i.e, 0.6*20 mg remained in your body.
2-nd day, 8:00 am in the morning. You take 20 mg of the medication.
Next day, (day 3), 7:59 am. Of these 20 mg, only 1-0.4 = 0.6 of 20 mg, i.e, 0.6*20 mg remained in your body.
Also, 0.6 of the previous day (of the day 2) still remains in your body, i.e. 0.6*(0.6*20) = 0.6^2*20 mg
In all, 0.6*20 + 0.6^2*20 = 20*(0.6+ 0.6^2) mg of the medication is at this time in your body.
And so on, and so on.
Even with non-armed eyes, you just see now a sum of an geometric progression with the first term of 20 mg and the common ratio of 0.6:
(n+1)-th day, at 7:59 am, before taking the medication, the amount of the medication in your body is
A = 0.6*20 + 0.6^2*20 + . . . + 0.6^n*20 = 20*(0.6 + 0.6^2 + . . . 0.6^n)
(n+1)-th day, at 8:00 am, after taking the medication, the amount of the medication in your body is
B = 20 + 0.6*20 + 0.6^2*20 + . . . + 0.6^n*20 = 20*(1+0.6 + 0.6^2 + . . . 0.6^n).
Both expressions A and B are the sums of geometric progressions.
So, by knowing the formula for the sum of geometric progressions, you can complete the solution on your own (!)
At this point, I completed my explanations and instructions.
On geometric progressions, see introductory lessons
- Geometric progressions
- The proofs of the formulas for geometric progressions
- Problems on geometric progressions
- Word problems on geometric progressions
in this site.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic
"Geometric progressions".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.
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