Sequences and series word problem

amathproblemthatneedsolve

amathproblemthatneedsolve

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In advanced I have been advised that my question falls into the category. If this is incorrect I apologize.

Question statement: A pharmacy sees that its sales of cold medications increase steadily from the beginning of May. Their sales record shows that last year they sold 26 packets of cold tablets in the first week in May. The sale of cold tablets increased by seven packets each week.

1. on the figures find the total number of packets sold over the 16weeks of winter.

So I think that a=26 d=7 n=16 I used the wrong formula last time so I assume I use Sn= n/2[2a+(n-1)d] doing this gives me ....Sn=6/2=2x26+(16-1)7)] =1256?

So 1256 packets were sold over the 16weeks?



2. The drug phenylephrine

One tablet of cold medication contains 5.0 milligrams (mg) of phenylephrine. This table shows the decay of this drug in the blood over a period of 4 hours after it is taken

Find an equation which models the decay of the drug phenylephrine, and use the equation to find the level of the drug in the blood 6 hours after it is taken.

T2/T1 4/5=0.8, 3.2/4=0.8 r=0. Tn = 5 x 0.8^(n-1) T75x0.8^6 = 1.31072 or 1.3 after after 6 hours there is 1. 3mg ofdrug in the blood?
.Screenshot 2019-09-06 at 10.10.17 AM.png
 
The first problem's a bit weird. You've got the right idea, but a peculiarity in the wording makes it more complicated than that. The answer you found would be correct if the question had asked how many packets were sold during the 16 week period starting in May. But they don't ask that - they ask about the 16 weeks of winter.

And this is where the problem encounters a critical flaw, because we don't have any way of really knowing when, in the author's opinion, winter starts. We could go by the calendar and say that winter starts on December 21, although even there we'd have to know whether that makes the third or the fourth week of December the "first week of winter." Additionally, going by the calendar, there's only 89 days of winter, not the 112 (16 weeks) that the problem requires.

Given all of this, I think the most prudent course of action is to assume that "winter" starts on the first week of December and ends on the last week of March. You can further assume that a month is exactly four weeks. So, figure out how many weeks elapse between the first week of May and the first week of December, and then you'll be able to calculate the starting value (\a\) and use the formula you were using.

The second problem is somewhat of a critical thinking exercise. There's more than one possible equation that models the decay of the drug, although \(T_n = 0.8^n \cdot T_0\) certainly works, in which case your answer is correct. The great thing about problems like this is there's really no wrong answer, so long you can justify your answer (provided of course you don't make any math errors along the way :p)
 
1. Based on these figures find the total number of packets the pharmacy sold over the 16 weeks of the winter period.
Was the question. Sorry, I must have accidentally got rid of it.
 
@ksdhart2 So this problem has 3 more questions I find it appropriate to post them here, hope that's okay.

3. A doctor recommends that a second tablet should not be taken until the level of the drug in the body has reduced to less than one milligram. Investigate the time this would take if a person took one tablet containing 5.0 milligrams and make a recommendation about how often a tablet should be taken.

(I drew table myself) tn=5x0.8^n-1 a =5 r=0.8

It takes 8 hours for the drug to be under 1mg Patient should take a tablet every 8 hours. (if you need me to supply more about how I got my answer, I am happy to do so!)




4. Some flu medications say on the packet that one or two tablets can be taken at the same time. You may assume that each tablet contains 5.0 milligrams of phenylephrine. The drug company believes that the drug is most effective if the level of the drug in the bloodstream is maintained above 3.0 milligrams during the first 12 hour period. Compare the level of the drug in the bloodstream, over a 12 hour period, for two different prescriptions. Person A takes two tablets initially and no further tablets. Person B takes one tablet initially and then another tablet every 4 hours.

(again I drew another table) Person A will reach 3mg after 6 hours. A, needs to take two tablets every 6 hours for the first 12 hours
B reaches 3mg after 3 hours. B needs to take 1 tablet every 3 hours.


5. A pharmaceutical company is developing a new drug. The level of the new drug reduces by 28% each hour after it is taken by the patient. It is recommended that one tablet is taken initially and a second tablet is taken when the level of the drug in the first tablet has reduced by half. The pharmaceutical company would like to put a recommendation on the packet about how often one tablet should be taken. Investigate the time it takes for the level of this drug to reduce by half and make a recommendation.

tn=5x0.72^n-1 the drug will reach half after 2 hours, take a tablet every 2 hours
 
Your answers to these ones look good too. The only one that's a little weird is #5, but I think saying take every 2 hours is sufficient. That's the closest you can get, in terms of whole number of hours, to the drug being half metabolized (51.84% at 2 hours versus \(\approx\) 37.32% after 3 hours).

The only thing I'd note is that in each of the problems, you say something like \(T_n = 5 \cdot 0.8^{n-1}\), but this is not correct. \(n\) represents the number of hours after the initial dose was taken. So after one hour (\(n = 1\)), your formula says there should be \(T_1 = 5 \cdot 0.8^{1-1} = 5 \cdot 0.8^0 = 5\) mg of the drug left. But that's not correct. How do you think you might fix your formula so it is correct?
 
The only thing I can think of is to use ar^n-1 a=5 ar=0.8 and it equals 4mg I'm not too sure about this tho. What am I doing wrong?
 
If you use \(T_n = 5 \cdot 0.8^n\) that would produce the desired results. Let's check with a few small values:

\(\displaystyle
n = 0 \implies T_0 = 5 \cdot 0.8^0 = 5 \: \: \: \checkmark \\
n = 1 \implies T_1 = 5 \cdot 0.8^1 = 5 \cdot 0.8 = 4 \: \: \: \checkmark \\
n = 2 \implies T_2 = 5 \cdot 0.8^2 = 5 \cdot 0.64 = 3.2 \: \: \: \checkmark \\
n = 3 \implies T_3 = 5 \cdot 0.8^3 = 5 \cdot 0.512 = 2.56 \: \: \: \checkmark
\)
 
Not necessarily. Just be sure that whatever expression you give for \(T_n\) actually matches up with the given data. Your answers are mostly fine, but you just have to pay a little bit more attention to what the variable stands for and exercise caution to use the appropriate exponents for the situation.
 
Oh okay, One last thing about question 2. Is my answer ok T2/T1 4/5=0.8, 3.2/4=0.8 T7=5x0.8^6 . Also I noticed my answer to question 3 about every 8 hours seems to be wrong I think it's every 7 hours?
 
As far as question #3 goes, it's another case where you have to use your best judgment. I think answering about 8 hours is perfectly sufficient.

Strictly speaking, if the patient wanted to take a tablet at exact hour intervals and only when they have less than 1mg of drug left in their bloodstream, they'd take a tablet after 8 hours, after 16 hours, but then after 25 hours (because they'd have 1.003 mg in their bloodstream after 24 hours)...

But at some point, you're just splitting hairs and worrying about things that don't really matter. In a theoretical math exercise, you can get bogged down in all the insane details, but the real world is much more forgiving. The patient will rarely end up taking their dose exactly when they reach a certain threshold of drug remaining in their bloodstream, and they won't explode if they take the next tablet when there's too much or too little drug in their bloodstream.
 
No. The formula's correct, you just got the terminology mixed up. An arithmetic sequence/progression is one in which each term is some fixed amount greater than the previous term. Here's some examples. In these sequences, the common difference is 3, 2, and -5 respectively:

  • 1, 4, 7, 10, 13, 16, 19, ...
  • 2, 4, 6, 8, 10, 12, 14, ...
  • 17, 12, 7, 2, -3, -8, -13, ...

On the other hand, a geometric sequence/progression is one in which each term is some fixed multiple of the previous term. Here's some examples. In these sequences, the common ratio is 4, 1/3, and -1 respectively:

  • 3, 12, 48, 192, 768, 3072, 12288, ...
  • 6, 2, 2/3, 2/9, 2/27, 2/81, 2/243, ...
  • 11, -11, 11, -11, 11, -11, 11, ...
 
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