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

1 Answer
Dec 5, 2015

Answer:

#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#