Assumptions:

Each vampire must suck the blood of one human per day. Each human who has their blood sucked by a vampire becomes a vampire.

Current world population: 7.1 * 1012

Let v n be the number of vampires in existence n days after the appearance of the first vampire(s).

Since each vampire turns one human per day, it follows that over the course of 24 hours, the population of vampires will increase by the value of v n at the start of the day in question, and the population of humans will decrease by the same amount. In other words:

v n+1 = 2v n

=> v n = 2v n-1

Proposition: v n = 2nv 0

Prove for n = 1:

v 1 = 21v 0

v 1 = 2v 0

which is true, as it satisfies the statment that v n = 2v n-1

Assume true for n = k:

v k = 2kv 0

Prove true for n = k + 1:

v k+1 = 2k+1v 0

v k+1 = 2k * 21 v 0

v k+1 = 2 (2kv 0 )

v k+1 = 2v k

which is true, as it satisfies v n+ 1 = 2v n .

Thus, we establish that, if the proposition is true when n takes any value k, it is also true when n takes the value k+1. We have also proven it to be true when n=1, thus it is true for all value of n.

When v n = 7.1 * 1012, all humans will have become vampires and thus humanity will be extinct. The value of n at this point gives the number of days humanity endured.

Let v 0 = 1.

v n = 7.1 * 1012

2nv 0 = 7.1 * 1012

2n = 7.1 * 1012

n = log 2 (7.1 * 1012 )

n = log 10 (7.1 * 1012 ) / log 10 2

n = 42.69

Thus, if a single vampire came into existence, all humanity would be exinct within 43 days. This also works for zombies, by the way – however, zombie movies tend to base their entire plots around this rapid expansion.

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