display | more...

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 vn be the number of vampires n days after the appearance of the first vampire. When vn = 6,782,504,562, humanity will be extinct.

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 vn. In other words:

vn+1 = vn

=>vn = 2vn-1

Proposition: vn = 2nv0

Prove true for n = 1

v1 = 21v0

v1 = 2v0, which is true, because it agrees with vn+1 = vn.

Assume true for n=k

vk = 2kv0

Prove true for n=k+1

vk+1 = 2k+1v0

vk+1 = 2vk

=>2vk = 2k21v0

=>vk = 2kv0, 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 vn = 2nv0

When vn ≥ 6,782,504,562, humanity will have gone extinct.

2nv0 = 6,782,504,562

Let v0 = 1.

2n(1) = 6,782,504,562

2n = 6,782,504,562

=>n = log26,782,504,562

=>n ≈ 3.322log106,782,504,562

=>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.

Log in or register to write something here or to contact authors.