Question 1201437: you and your parents are working on a scrapbook representing your family tree. He would like to include one page for the biography of each and sister, and your parents want you to include a page on yourself, classify the number of ancestors in each generation, as an arithmetic or geometric sequence.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52788)   (Show Source):
You can put this solution on YOUR website! .
It is difficult to relate in the mind, how the content of the problem is connected with the question.
Not very smooth reading.
Math problem, written in a right way, should go more smoothly.
The number of your ancestors in the n-th generation before you is  , n = 1, 2, 3, . . .
2, 4, 8 and so on, making a geometric progression with the common ratio of 2.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Answer: Geometric sequence
2^n ancestors in generation n
n = 1 is the previous generation, compared to your generation
n = 2 is the previous two generations, etc
Explanation:
Refer to this diagram to see a template of a simplified family tree
The bottom of the family tree represents the present day and current generation.
The youngest persons are always at the bottom, with the oldest at the top.
As you move up the page, you move back in time.
You can think of it like time flowing downhill, so going up the page reverses that flow.
As we move back through each generation, we double the number of people.
This is simply from the obvious fact two parents are needed for any given individual.
Therefore, we have a geometric sequence with common ratio r = 2.
We start with one person, then double to 2, that doubles to 4, and so on.
So that's how we end up with 2^n, where n is the generation number mentioned earlier.
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Interesting fact about powers of 2:
Let's say we add you and your parents to get 1+2 = 3 people
Now add on the four grand parents: 3+4 = 7
Then the 8 great grand parents: 7+8 = 15
Then the 16 great great grand parents: 15+16 = 31
The subtotals were:
3, 7, 15, 31
Add 1 to each of those
3+1 = 4
7+1 = 8
15+1 = 16
31+1 = 32
We have powers of 2.
Rule: the sum of the terms 1+2+4+8+...+2^n is 2^(n+1)-1
I'll let you explore sums. Try out something like going back n = 6 generations to see how many total family members there are.
Keep in mind that this is a simplified family tree. By that I mean we make the assumption that each parenting couple has one child only.
Realistically a family could have multiple children which would of course complicate things greatly. It would also break the geometric sequence property where the common ratio was r = 2. This is probably why your teacher will want the more simplified family version.
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