Have you ever looked through old family pictures and constructed yourself, or looked at, a family tree? In this activity you will explore some mathematics related to family trees, and also the (sometimes tricky) question of how to properly apply mathematics to real-world situations.
I. For the purpose of the following questions, we will ignore divorces, second marriages, adoptions, etc. By ``how many great-grandparents did you have?'' we mean, ``while they were alive''
i. As a power of two, how many direct ancestors did you have 7 generations ago? __________
ii. How many direct ancestors did each of them have, 3 generations before them?_______(that is, as a power of 2, each of the ancestors in i. had ____ ancestors which came 3 generations before them).
iii. The product of the two numbers above must equal the number of direct ancestors you have coming from 10 generations ago. But the number of "10 generations ago" ancestors you have is (as a power of 2) equal to _______
iv. Write down the two mathematical expressions which must equal; write an equation with terms on each side: _____________
v. What is an expression for the number of ancestors i generations ago? __________ . Using the fact that each of these ancestors had, j generations before them, _______ ancestors, and that the product of the previous two numbers equal the number of ancestors you have from i+j generations ago, what equation to you get? _____________