Question

A patient is prescribed 50mg of medication to be taken every morning for the rest of their life "forever". The physiology of the medication is such that the body metabolizes it so that at the end of a 24 hour period there is only 10% of the medication left in the body. So, on the first morning (day 0) the patient takes 50mg and has 50mgs in their body. the next morning (day 1) there is 5mg left and they take another 50mg. At that moment there is 55mg in their body. The next morning (day 2) there is 5.5 mg left in their body and another 50 mg dose is taken giving 55.5 mg in the body.

Answer 1

QUESTION:
As the days "go to infinity" what is the maximum amount of medication that will ever be in the patients body?

Answer 2

I've worked out this formula:\[\sum_{d=0}^{\infty}50(1/10)^{d}\]

Answer 3

And am pretty sure it is describing the phenomenon: http://www.wolframalpha.com/input/?i=+sum+50%281%2F10%29^d%2C+d%3D0+to+0

Answer 4

hmmim thinkin it has to b at least 55. cuz at the end ur gonna end up with 55 and5/9 but i havent worked out the formula yet

Answer 5

55.555 is d=4 in the series

Answer 6

ur series looks correct, i just dont like the answer wolframalpha gave u

Answer 7

Th sum at least

Answer 8

Its the sum

Answer 9

o i kno u ddnt put the summation to infinity. u left it at zero

Answer 10

So at d=0 we have: 50(1/10)^0 -> 50(1) = 1

Answer 11

ah yea

Answer 12

If I put infinity in it says \[\approx 55.556\]

Answer 13

so at the end u should get 55.5repeating which is 55 +5/9

Answer 14

I wanna know why though

Answer 15

Do I have to take a limit in some way?

Answer 16

well u solved it by doing a summation so u would just have to solve ur summation

Answer 17

Its weird though

Answer 18

Wouldnt the amount just keep increasing by a tiny .5ish amount?

Answer 19

actually each day the amount increases by 5/(10^n). so each day u would add a 5 after the last 1 so the 5 would just grow longer and longer into infinity.
55.5
55.55
55.555
55.5555
55.55555 and so on

Answer 20

So wouldnt the max amount be 55.56?

Answer 21

If you are going to approximate anyway on 55.5555556 like wolfram does

Answer 22

Since I am looking for the max it is safe to just say 55.56?

Answer 23

where are u writing ur answer. the max amount is 500/9 if u can write that. u cant really write the max amount as a decimal

Answer 24

Why is that the max amount though?

Answer 25

It seems arbitrary to me...

Answer 26

Is there some method you are doin to get that exact amount?

Answer 27

i solved ur summation. (1/10)^d is an r series

Answer 28

I guess I dont know how to solve summations?

Answer 29

if u solve it, it becomes 10/9 and then u multiply that times 50 so u get 500/9

Answer 30

do u kno the r series?

Answer 31

no

Answer 32

I have learned test to determine convergence or divergence and am learning power series.

Answer 33

summation from o to infinity of r^n=r/(1-r) if 0<r<1

Answer 34

Not familiar

Answer 35

geometric series!! u should have learnt that before u learned tests for convergence and divergence

Answer 36

Summer class lol

Answer 37

I am looking up "how to solve an infinite summation"

Answer 38

try looking up geometric series

Answer 39

Is it very involved or just a few steps?

Answer 40

its actually quite simple. just plug in the numbers

Answer 41

\[\sum_{o}^{\infty}r=r/(1-r)\]

Answer 42

but it only works if:
0<r<1

Answer 43

Ok thanks for the help!

Answer 44

np