Recurrence relations: NYJC 2024 Promo Q4

Nanyang Junior College2024 PromoPromo Q4●●○ Standard6 marks

What this question tests

Recurrence relations.

Question

During a clinical trial, the concentration $U$ , 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 $U_3 = 88, \qquad U_4 = 76 \quad\text{and}\quad U_5 = 70,$ where $U_t$ denotes the reading $t$ hours after the drug was first administered.

It is believed that $U$ satisfies the relationship $U_{t+1} = p + qU_t, \qquad t \geq 0,$ 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]

Show full worked solution▾

(a)

$\begin{aligned} U_4=p+qU_3&\Rightarrow 76=p+88q \text{ --- (1)}\\ U_5=p+qU_4&\Rightarrow 70=p+76q \text{ --- (2)} \end{aligned}$ Solving equation (1) and (2): $p=32,\quad q=0.5$

(b)

Method 1:

$\begin{aligned} U_3=p+qU_2&\Rightarrow 88=32+(0.5)U_2\Rightarrow U_2=112\\ U_2=p+qU_1&\Rightarrow 112=32+(0.5)U_1\Rightarrow U_1=160\\ U_1=p+qU_0&\Rightarrow 160=32+(0.5)U_0\Rightarrow U_0=256 \end{aligned}$

Method 2:

$\begin{aligned} U_3&=32+\tfrac12U_2\\ &=32+\tfrac12(32+\tfrac12U_1)\\ &=32+\tfrac12\left(32+\tfrac12(32+\tfrac12U_0)\right)\\ 88&=32+\tfrac12\left(32+\tfrac12(32+\tfrac12U_0)\right)\\ \therefore U_0&=256 \end{aligned}$

(c)

Method 1: Using GC to get 64

$Nanyang Junior College 2024 Promo Q4: solution diagram$

Method 2:

As $t\to\infty$ , $U_t\to L$ , $U_{t+1}\to L$ $\begin{aligned} L&=32+\tfrac12L\\2L&=64+L\\L&=64 \end{aligned}$ The readings decreases and converges to 64.

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