# A person has two parents, four grandparents, eight great-grandparents, and so on. How many ancestors does a person have 15 generations back?

Dec 5, 2015

${2}^{15} = 32768$

#### Explanation:

Assuming the ancestors are distinct (very unlikely), each generation is double the size of the following generation. So $15$ generations back will be ${2}^{15} = 32768$ ancestors.

The total number of ancestors in all generations back $15$ generations will be ${2}^{16} - 2 = 65534$ (not counting the person themselves).

The sequence:

$1 , 2 , 4 , 8 , 16 , 32 , \ldots , 32768$

is a geometric sequence with common ratio $2$.

The sum of the first $N$ terms of a geometric sequence with general term ${a}_{n} = a {r}^{n - 1}$ is:

$\frac{a \left({r}^{N + 1} - 1\right)}{r - 1}$

since:

$\left(r - 1\right) {\sum}_{n = 1}^{N} a {r}^{n - 1}$

$= r {\sum}_{n = 1}^{N} a {r}^{n - 1} - {\sum}_{n = 1}^{N} a {r}^{n - 1}$

$= {\sum}_{n = 2}^{N + 1} a {r}^{n - 1} - {\sum}_{n = 1}^{N} a {r}^{n - 1}$

$= \left(a {r}^{N + 1} + \textcolor{red}{\cancel{\textcolor{b l a c k}{{\sum}_{n = 2}^{N} a {r}^{n - 1}}}}\right) - \left(a + \textcolor{red}{\cancel{\textcolor{b l a c k}{{\sum}_{n = 2}^{N} a {r}^{n - 1}}}}\right)$

$= a \left({r}^{N + 1} - 1\right)$

Hence $1 + 2 + 4 + \ldots + {2}^{15} = {2}^{16} - 1 = 65535$