Mastering Mathematics Smartly
by Wee Wen Shih

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Establishing recurrences

Sometimes, questions may be asked requiring students to formulate recurrence relations based on the information provided. We shall look at two examples below.

Example 1: A certain infectious disease spreads rapidly. On day 1, 3 people are infected. On day 2, 9 people have the disease. On day 3, there were 27 infectants. Assuming a constant rate of infection, write down an appropriate recurrence relation to describe this context.

Approach:
Let eq=u_n represent the number of cases of infection on day n.
 
On day 1, 3 people are infected, so we have eq=u_1 = 3 as our base case. 

For each subsequent day, the number of cases is tripled, so we obtain eq=u_n = 3u_{n - 1} where eq=n \ge 2.

Note that the resulting sequence follows a geometric progression, since eq=\frac{u_n}{u_{n-1}}=3.

Example 2: We continue from example 1. To slow down the spread of this disease, the health authority proposes a measure to quarantine 15 patients each day starting from the third day. Assuming a constant rate of infection, write down an appropriate recurrence relation to describe this context.

Approach:
Let eq=w_n represent the number of non-quarantined cases of infection on day n.

The base cases are eq=w_1 = 3, eq=w_2 = 9, eq=w_3 = 27-15=12 (because the quarantine measure kicks in only on day 3).

For each subsequent day, the number of cases is again tripled and 15 people are to be taken out to be quarantined, so we have eq=w_n = 3w_{n - 1} - 15 where eq=n \ge 4.

As an exercise, I leave this question for the keen learner to reflect upon:
Show that the number of non-quarantined cases of infection on the Nth day of the measure is eq=\frac{1}{2}\left(3^{N+1} + 15\right). Hence deduce the long-term behaviour of the disease.

Hints: Read my first article on recurrences. What happens if eq=N \rightarrow \infty.