Problems in Context

Video lesson · from 19:351 worked example

One lesson video covers all of Recurrence Relations, so it opens at 19:35 for this topic — not from the beginning.

Theory

Recurrence relations are most often examined through a real situation — a population, a drug dose, a bank balance, a pollutant in a loch. The mathematics is exactly the same as before; the difficulty is turning the words into $u_{n+1} = au_n + b$ .

Read the description for two separate things:

a proportional change — a percentage rise or fall, which becomes the multiplier $a$ ;
a fixed change — a quantity added or removed each period, which becomes $b$ .

A fall of 15% leaves 85%, so $a = 0.85$ ; a rise of 15% gives $a = 1.15$ . A quantity removed each time makes $b$ negative.

The long-term behaviour question — “will the level ever exceed…”, “what happens in the long run” — is asking for the limit, which requires $-1 \lt a \lt 1$ .

The Golden Rule: always say what $u_n$ stands for, including its units and the period it covers. A recurrence relation with undefined terms does not answer a context question.

⚠️ Common Examiner Traps

Translate the words carefully: “increases by 5% then 200 are removed” gives $u_{n+1} = 1.05u_n - 200$ . Order and sign both matter.
Answer the question that was asked: often the useful answer is whether a level is safe or sustainable, not the value of $u_{10}$ . Read to the end.
Compare with the limit to justify: a long-term claim needs the limit and the condition $-1 \lt a \lt 1$ , not a list of terms.
Watch which term the question wants: “after 4 years” may be $u_4$ or $u_3$ depending on where the sequence starts.

Worked examples

Example 1

A bank has a sales drive in order to attract new customers. Every 6 month period, the bank estimates that it loses 1.7% of its customers to competitors but attracts 8000 new customers. It has 0.5 million customers at the start of its sales drive.

a) How many customers will the bank have in 4 years?
b) What will happen to the number of customers in the long term if the current sales drive continued?
c) How many new customers would the bank have to attract in each 6 month period to maintain 0.5 million customers?

a) Recurrence relation: $U_{n+1} = 0.983U_n + 8000$ where $U_0 = 500{,}000$ .

4 years means 8 periods of 6 months ( $n = 8$ ).

U_1 = 0.983(500{,}000) + 8000 = 499{,}500

\vdots

U_8 \approx 496{,}252.17

The bank will have approximately 496,252 customers.

b) Long term limit:

\begin{aligned} L &= \frac{8000}{1 - 0.983} \\ &= \frac{8000}{0.017} \\ &\approx 470{,}588 \end{aligned}

The number of customers will fall and settle at approximately 470,588.

c) To maintain 0.5 million customers, the limit must be 500,000.

\begin{aligned} 500{,}000 &= \frac{b}{1 - 0.983} \\ 500{,}000 &= \frac{b}{0.017} \\ b &= 500{,}000 \times 0.017 \\ &= 8500 \end{aligned}

The bank would need to attract 8500 new customers each 6 month period.