Homework 1
Due Date: Friday, September 2
Instructions: Feel free to work together with one or two other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with. You can turn in your homework assignment by e-mailing me your solutions.
- The sequence
\[
1,\quad 1,\quad 7,\quad 13,\quad 55,\quad 133,\quad 463,\quad \ldots
\]
obeys the recurrence relation
\[
a_n \,=\, a_{n-1} + 6a_{n-2},
\]
where \(a_1=a_2=1\). Use the method described at the end of Section 1.1 to find an explicit formula for \(a_n\) in terms of \(n\).
- A disease is spreading through a large population. Once a person becomes infected, the disease has a latency period of 1-2 days, after which the person is contagious. A contagious individual typically infects 3-4 other people per day for a period of several weeks.
- Construct a mathematical model for the spread of this disease using a linear recurrence relation. Justify your answer by explaining the connection between your equation and the characteristics of the disease.
- Suppose that an outbreak of the disease begins when a single infected person becomes contagious. Estimate the number of infected individuals after one week. Explain your reasoning.