Everyone knows that vampires are hungry, immortal monsters. What if they turned a new victim into a vampire every week? Because of exponential growth, they would quickly run out of victims, not just in the local area, but on the face of the Earth! This must mean that vampires do not exist.

Of course, there are real contagious, fatal diseases in the world. Unlike with vampirism, some people are immune, some people can be cured, and some people die from these diseases. Students will analyze the consequences of a more realistic contagion and discuss factors that keep a contagion from spreading among a population.

### Students will

• Write and solve an exponential equation in one variable to determine how many weeks before the entire human race becomes vampires
• Prove informally by contradiction that vampires of this type do not exist
• Model the number of victims of a hypothetical infectious disease with a geometric sequence and/or series
• By setting the sum of a finite geometric series equal to a value and solving, find the number of weeks until the total number of victims reaches the world population
• Adjust the model to account for a certain percentage of the population being immune to infection
• Discuss the factors that prevent deadly, contagious diseases from infecting too many people

### Before you begin

The more difficult questions in this lesson require using the formula for the sum of a finite geometric series or a spreadsheet to perform the same calculations recursively. This lesson provides an opportunity for students who have learned about the formula for the sum of a finite geometric series to apply it.