Sequences/Series Problem Based on Medication

[SammyS]

For the 3% remaining problem I did this:

T=3.921 pills remaining, so

.03=3.921(1/2^{t/1 day}=3.921^t

log(1/2)(t)=log(.03/3.921)=log(.007)
log(1/2)(t)=-2.1163
-.301t=-2.1163
t=7.031 days (round up?)

Is this moderately close? At least tell me that ;'(

I get that T = 3.921875 pills ≈ 3.922 pills , so you're probably close enough for that answer.

After, the 8th dose, the patient takes no more pills. The half-life of the pills is approximately 24 hours.
.03=3.921((1/2)^t) is correct for t measured in days, but 3.921(1/2)^t ≠ 3.921^t . (Probably a typo, b/c the result is very good.)

Yes, I concur with your answer..

 

[Ray Vickson]

For the 3% remaining problem I did this:

T=3.921 pills remaining, so

.03=3.921(1/2^{t/1 day}=3.921^t

log(1/2)(t)=log(.03/3.921)=log(.007)
log(1/2)(t)=-2.1163
-.301t=-2.1163
t=7.031 days (round up?)

Is this moderately close? At least tell me that ;'(

How did you get T = 3.921? We have T = 5 + 10*x + 15*x^2 + ... + 40*x^7, where x = 1/2. Just taking the first two terms already gives you 10, and the other terms will give you still more. The rest of you calculation makes no sense either. Why would you say that 3.921*1/2^t is the same as 3.921^t? (It isn't.) Do you honestly believe that, or are you just writing down random things out of desperation?

RGV

 

[SammyS]

How did you get T = 3.921? We have T = 5 + 10*x + 15*x^2 + ... + 40*x^7, where x = 1/2. Just taking the first two terms already gives you 10, and the other terms will give you still more. The rest of you calculation makes no sense either. Why would you say that 3.921*1/2^t is the same as 3.921^t? (It isn't.) Do you honestly believe that, or are you just writing down random things out of desperation?

RGV

Ray, T is measured in "number of medication tablets", so it's 1/5 of what you have.

The line after the one with 3.921^t, indicates that 3.921^t was a typo, or was ignored in the next step.

 

[SammyS]

mundane,

Have you done part (d) ?

A patient is prescribed n tablets of medication the first day, n-1 the second, and one tablet fewer each day until all tablets are gone. Write a formula that represents T_n , the number of medication tablets in the body right after taking all tablets. Find a closed form for T_n .​

 

[mundane]

Awesome, thanks, SammyS!

For T_n, in part d) the equation I have is nx^n-1+(n-1)x^n-2 ... + but I don't know what to end it with. Do I call the number of days y and incorporate that in? I am trying to make it summation notation but can't get anywhere,

 

[SammyS]

Awesome, thanks, SammyS!

For T_n, in part d) the equation I have is nx^n-1+(n-1)x^n-2 ... + but I don't know what to end it with. Do I call the number of days y and incorporate that in? I am trying to make it summation notation but can't get anywhere,

I assume that there would be enough (actually that's (n(n+1))/2 ) so that on the n^th day, the patient would take 1 tablet.

So, I would still end with ... + 4x³ + 3x² + 2x + 1 .

 

[mundane]

I assume that there would be enough (actually that's (n(n+1))/2 ) so that on the n^th day, the patient would take 1 tablet.

So, I would still end with ... + 4x³ + 3x² + 2x + 1 .

I am kind of confused just because in this case it seems like n should be shown as the amount of tablets for part d.

Is that closed form that you've written above? It seems like a subjective term so I wasn't sure.

 

[mundane]

I guess What I am trying to do is write it in summation notation for part d, but I don't know how to relate the last day's amount of n tablets. The only thing I can think of would be something (n-i

 

[mundane]

I guess What I am trying to do is write it in summation notation for part d, but I don't know how to relate the last day's amount of n tablets. The only thing I can think of would be something (n-n+1)x^(n-n) if n is the initial amount of tablets and but I don't know how to sum that up properly...

 

[SammyS]

I am kind of confused just because in this case it seems like n should be shown as the amount of tablets for part d.

Is that closed form that you've written above? It seems like a subjective term so I wasn't sure.

I guess What I am trying to do is write it in summation notation for part d, but I don't know how to relate the last day's amount of n tablets. The only thing I can think of would be something (n-i

For parts (a), (b), and (c), you start with 8 tablets, then 7, then 6, ... then 2, then 1, for a total of 36 tablets.

If for part (d), the total number of tablets is not some value that gives you a final day dosage that follows the pattern then you have a real mess. In fact, even if you follow the pattern on the last day, if that dosage is not a single tablet, then with the information given you are left hanging. You are not told the total number of tablets.

So, I suggest again the there are enough total tablets prescribed so that on the last day (the n^th day) the patient is given just one tablet.

That gives the patient n days worth of pills. The fact that it is suggested to use a subscript of n with the T, as in T_n, also suggests to me that the patent is given tablets for n days.

 

[mundane]

For parts (a), (b), and (c), you start with 8 tablets, then 7, then 6, ... then 2, then 1, for a total of 36 tablets.

If for part (d), the total number of tablets is not some value that gives you a final day dosage that follows the pattern then you have a real mess. In fact, even if you follow the pattern on the last day, if that dosage is not a single tablet, then with the information given you are left hanging. You are not told the total number of tablets.

So, I suggest again the there are enough total tablets prescribed so that on the last day (the n^th day) the patient is given just one tablet.

That gives the patient n days worth of pills. The fact that it is suggested to use a subscript of n with the T, as in T_n, also suggests to me that the patent is given tablets for n days.

I definitely understan what you mean... But in the previous context would that mean that n=8 because there are 8 days? It seems like part d would be asking the same question as b then, right? Maybe I'm way off.

My teacher said that at part d) n should equal the initial number of tablets for Amy amount of days, say 25 for example, and we should make a closed form for how many pills in relation to # of days are left in the body at the last dose of 1 pill.

 

[SammyS]

...

d) A patient is prescribed n tablets of medication the first day, n-1 the second, and one tablet fewer each day until all tablets are gone. Write a formula that represent T(sub)n, the number of medication tablets in the body right after taking all tablets. Find a closed form for T(sub)n."

The above is a direct quote from your Original Post in this thread.

I definitely understand what you mean... But in the previous context would that mean that n=8 because there are 8 days? It seems like part d would be asking the same question as b then, right? Maybe I'm way off.

My teacher said that at part d) n should equal the initial number of tablets for Any amount of days, say 25 for example, and we should make a closed form for how many pills in relation to # of days are left in the body at the last dose of 1 pill.

Well, if n pills (tablets) are taken the first day (" n should equal the initial number of tablets for any amount of days ") and pills are taken every day for say, 25 days (for example), then if we are to diminish the dose by one pill each day, until the 25th day, when only one pill is taken, --- if all that is true, then the number of pills taken the first day, n, must be 25, for this example. Right?

This is what I stated previously. --- or what I intended ti state.

It is very much like the case of taking 8 pills the first day, etc. in part (b), but it is different. In part (d), n pills are taken the 1st day, one fewer is taken each subsequent day, until the n^th day when one pill is taken. It's a generalization of part (b). Once you get your result, plugging in 8 for n, should give a result equivalent to the one obtained previously.

 

[mundane]

The above is a direct quote from your Original Post in this thread.


Well, if n pills (tablets) are taken the first day (" n should equal the initial number of tablets for any amount of days ") and pills are taken every day for say, 25 days (for example), then if we are to diminish the dose by one pill each day, until the 25th day, when only one pill is taken, --- if all that is true, then the number of pills taken the first day, n, must be 25, for this example. Right?

This is what I stated previously. --- or what I intended ti state.

It is very much like the case of taking 8 pills the first day, etc. in part (b), but it is different. In part (d), n pills are taken the 1st day, one fewer is taken each subsequent day, until the n^th day when one pill is taken. It's a generalization of part (b). Once you get your result, plugging in 8 for n, should give a result equivalent to the one obtained previously.

Ohhh, so can I integrate it like you showed me for part b? And that closed form would show how to get T for any number of days?

 

[mundane]

The one thing I don't get is how to express the last days. Thats why I asked about saying (n-i) Tablets since I didn't have a way to express the number of days, I guess.

 

[SammyS]

The one thing I don't get is how to express the last days. That's why I asked about saying (n-i) Tablets since I didn't have a way to express the number of days, I guess.

...

My teacher said that at part d) n should equal the initial number of tablets for Amy amount of days, say 25 for example, and we should make a closed form for how many pills in relation to # of days are left in the body at the last dose of 1 pill.

... at the last dose of one pill should answer your question above, in the first of the two quotes.

In post #42, I have given a fairly detailed version of my understanding of part (d). It's consistent with what your teacher said, and with the initial statement of the problem. Also, it's a more detailed version of my posts #34, #36 & #40, but with more detail and more complete.

To say it again, it appears to me that whatever number, n, of pills are taken the first day, there is a sufficient number of pills to follow the complete pattern & take one pill on day number n .

I can't make it any clearer than that.

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The steps in solving part (d) should be similar to the steps in solving part(b) .