Question 1205433: You and your parents are working on a scrap book representing your family tree. You would like to include one page for each ancestor, and your parents want you to include a page or yourself.
How many generations can you include with keeping it fewer than 300 pages, how many pages would you need to 12 generations of ancestors, and assuming that one generation spans about 30 years, what s the earliest year that will appear in the book?
Answer by ikleyn(52810)   (Show Source):
You can put this solution on YOUR website! .
You and your parents are working on a scrap book representing your family tree.
You would like to include one page for each ancestor, and your parents want you to include a page or yourself.
(a) How many generations can you include with keeping it fewer than 300 pages,
(b) how many pages would you need to 12 generations of ancestors,
(c) and assuming that one generation spans about 30 years, what s the earliest year that will appear in the book?
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In this family tree, you need 1 page for yourself; 2 pages for your parents,
4 pages for your grandparents, and so on:  pages for all ancestors of the n-th generation up from you.
So, for all ancestors <= N generations up from you (including you) you need
1 + 2 + 4 + 8 + . . . +  =  pages in your book.
(a) Having fever than 300 pages, you can place the info about N generations up from you, where
< 300, of  <= 300, N+1 <=  ~ 8.22.
So, to evaluate the integer number N of generations up, you need take N ~ 8.22-1 = 7.22
and round it to the closest lesser integer, which is 7.
Thus the ANSWER to question (a) is N = 7, or seven generations up from you,
counting you as the 0-th generation.
Below is the Table, counting pages
n the number of accessors the number of ancestors
in the n-th generation in all n generations
up from you including you up from you including you
(same as the number of pages)
----------------------------------------------------------------------------
0 (you) 1 1
1 2 1+2 = 3
2 4 1+2+4 = 7
3 8 7+8 = 15
4 16 15+16 = 31
5 32 31+32 = 63
6 64 63+64 = 127
7 128 127+128 = 255
Here, at the generation 7, you must stop, because you have no more than 299 pages.
(b) For 12 generations of ancestors, you need  = 8191 pages (including you).
(c) To answer (c), I should know exactly, for what number of generations this question does relate,
which is not given in the question.
Solved.
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