You have two biological parents, four biological grandparents, eight biological grandparents, and so on... Assuming that you are counting generations backward in time starting at generation 1 for your biological parents, and assuming that your biological ancestors are all different, define the sequence a_n to be the number of your biological ancestors at generation n. (so a₁=2, a₂=4, and so on)

1. Define the sequence a_n recursively. (i.e. provide initial conditions and a recurrence relation for a_n)
2. What type of sequence is a_n?
3. Give an analytical solution for a_n
4. Give a formula for A(n)= the total number of your biological ancestors for the past n generations. Explain how you derive your answer mathematically a_n and known formulas.
5. What would be the total number of your biological ancestors for the past 40 generations?
6. Assuming that each generation represents 25 years, how long is 40 generations? Explain your answer.
7. The total number of people who have ever lived is approximately 100 billion, which equals 10¹¹. Compare this fact with the answer to the previous question. What do you deduce? Explain your answer carefully.