Proposition: If one vampire were to appear on Earth, humanity would go extinct too fast for a stable equilibrium to develop.

Assumption 1: Each vampire must drain the blood of one human every 24 hours.

Assumption 2: A human who has had their blood sucked by a vampire will become a vampire.

Current population of Earth: 6,782,504,562 (Source; checked 00:31 GMT on September 7th, 2009)

Let *v _{n}* be the number of vampires

*n*days after the appearance of the first vampire. When

*v*= 6,782,504,562, humanity will be extinct.

_{n}If each vampire sucks the blood of one human in a 24-hour period, then in each 24-hour period, the population of vampires will increase by *v _{n}*. In other words:

*v*_{n+1} = *v _{n}*

=>*v _{n}* = 2

*v*

_{n-1}

Proposition: *v _{n}* = 2

^{n}

*v*

_{0}

Prove true for *n* = 1

*v*_{1} = 2^{1}*v*_{0}

*v*_{1} = 2*v*_{0}, which is true, because it agrees with *v*_{n+1} = *v*_{n}.

Assume true for *n=k*

*v _{k}* = 2

^{k}

*v*

_{0}

Prove true for *n=k*+1

*v*_{k+1} = 2^{k+1}*v*_{0}

*v*_{k+1} = 2*v*_{k}

=>2*v*_{k} = 2^{k}2^{1}*v*_{0}

=>*v*_{k} = 2^{k}*v*_{0}, which has already been assumed to be true.

Thus, for all *k*, the equation is true for *k*+1.

=> for all *n*, the equation is true for *n*+1. Since it has been proven true for *n*=1, it follows that the equation is true for all values of *n*.

Thus, it has been proved by induction that *v _{n}* = 2

^{n}

*v*

_{0}

When *v _{n}* ≥ 6,782,504,562, humanity will have gone extinct.

2^{n}*v*_{0} = 6,782,504,562

Let *v*_{0} = 1.

2^{n}(1) = 6,782,504,562

2^{n} = 6,782,504,562

=>*n* = log_{2}6,782,504,562

=>*n* ≈ 3.322log_{10}6,782,504,562

=>*n* ≈ 32.66 days.

Thus, if you were to start with a single vampire, humanity would be extinct within 33 days. That's why authors should only have their vampires drink a little bit of blood, every so often.