vampirism is mathematically impossible (essay)

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_{n = 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 = 2^kv0}

_{Prove true for n=k+1}

_{vk+1 = 2^k+1v0}

_{vk+1 = 2vk}

_{=>2vk = 2^k2¹v0}

_{=>vk = 2^kv0, 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.}