During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at on

Question

During a clinical trial, the concentration ${{U}_{t}}$ of a specific blood agent is measured at one-hour intervals following the initial administration of the trial drug to a patient.

The following readings were obtained:

${{U}_{3}}=88$, ${{U}_{4}}=76$ and ${{U}_{5}}=70$,

where ${{U}_{t}}$ denotes the reading $t$ hours after the drug was first administered.

It is believed that $U$ satisfies the relationship

${{U}_{t+1}}=p+q{{U}_{t}}$, $t\ge 0$,

where $p$ and $q$ are constants.

Find the values of $p$ and $q$.[2]
Determine the initial concentration of the blood agent when the drug was initially administered.[3]
Describe how the concentration behaves over a long period of time.[1]

Accepted Answer

(a) $p = 32$, $q = \frac{1}{2}$; (b) $U_0 = 256$; (c) $L=64$. The readings decrease and converge to $64$.

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