Number of Ancestors 1000 Years Ago

Date: 11/22/1999 at 00:02:42
From: Jana Larman
Subject: Anthropology

Assuming 25 years per generation, calculate how many living relatives 
you would have had 1,000 years ago (living in AD 998 - not the 
cumulative number). What is the simplified formula for this type of 
calculation?

Date: 11/22/1999 at 10:56:54
From: Doctor Rob
Subject: Re: Anthropology

Thanks for writing to Ask Dr. Math, Jana.

Boy, did you ask the right guy - not only do I know the answer, I am a 
genealogist, too.

1 generation ago,    25 years ago, you had  2  parents
2 generations ago,   50 years ago, you had  4  grandparents
3 generations ago,   75 years ago, you had  8  great-grandparents
N generations ago, 25*N years ago, you had 2^N great-...-grandparents

For 1000 years ago set 25*N = 1000, solve for N, and raise 2 to that 
power.

For Y years ago, set 25*N = Y, solve for N, and raise 2 to that power.

As a check, my answer came out to more than 1,000,000,000,000.

This is simplified, because we are not accounting for overlap of 
lifespans, so that members of two separate generations could be living 
at the same time. This also ignores the fact that cousins marry, and, 
in fact, such marriages are rather more common than generally thought. 
For example, in my 11th generation of ancestors, one couple appears no 
fewer than five times! Many other ancestors appear more than once, 
too, so instead of 2048 = 2^11 different people, a little under 2000 
actually appear. This phenomenon gets more pronounced the farther back 
one goes, so a smaller and smaller proportion of one's family tree 
consists of distinct people with each generation.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/