Question

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

Answer 1

`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,...,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:

`(a(r^(N+1)-1))/(r-1)`

since:

`(r-1) 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)`

`= (a r^(N+1) + color(red)(cancel(color(black)(sum_(n=2)^N a r^(n-1))))) - (a + color(red)(cancel(color(black)(sum_(n=2)^N a r^(n-1)))))`

`= a (r^(N+1) - 1)`

Hence `1+2+4+...+2^15 = 2^16-1 = 65535`