Recurrence relations

Just wondering if there's an equation of how to find the lower limit?

Question is:

3. A patient is given an initial life saving dose of 100mg of a drug. As the effect of the drug
wears off it is topped up hourly according to the recurrence relation:

Un+1=aUn+b

After the first top-up there is 90mg of the drug in the patient’s body. After 2 top – ups there is 84mg.

(a) Use this information to calculate the values of a and b. a=0.6 and b= 30

(b) This treatment is deemed effective if, in the long term, the strength of the drug
in the body doesn’t fall below 46mg, and safe, if the strength is not consistently
above 80mg.

Is this treatment both effective and safe in the long term?
You must show working to justify your answer.

(edited 11 years ago)

Reply 1

the bear

your image did not appear

Reply 2

Katie McArdle

Original post by the bear

your image did not appear

It was the recurrence relation formula, thanks :smile:

Reply 3

the bear

to find the long term value of the dose you say

U = aU + b

because the new dose is now the same as the old dose

rearrange to find U

Reply 4

Katie McArdle

Original post by the bear

to find the long term value of the dose you say

U = aU + b

because the new dose is now the same as the old dose

rearrange to find U

I got the 75mg, how do i get the lower limit of 45mg?

Reply 5

the bear

putting the initial dose as 100

you get a nested expression for subsequent doses:

0.6(0.6(0.6(0.6(0.6(0.6*100 + 30)+30)+30)+30)+30) + 30

now examine the bracketed expressions beginning with the innermost... it is over 30, so when you * by 0.6 it is over 18...but then you add 30 which takes it to 48

the same process is repeated, multiplying a number over 30 by 0.6 and adding 30... so all the doses are over 48