MATH SOLVE

2 months ago

Q:
# You have two parents, four grandparents, eight greatgrandparents, and so forth.a. if all your ancestors were distinct, what would be the total number of your ancestors for the past 40 generations (counting your parents' generation as number one)? (hint: use the formula for the sum of a geometric sequence.)b. assuming that each generation represents 25 years, how long is 40 generations?c. the total number of people who have ever lived is approximately 10 billion, which equals 10 10 people. compare this fact with the answer to part (a). what do you deduce?

Accepted Solution

A:

a.

In the first generation we have 2 ancestors.

In the second generation we have 4 ancestors or [tex]2^{2} [/tex] ancestors.

In the third generation we have 8 ancestors ot [tex]2^{3} [/tex] ancestors.

We can see that each succesive generation has two times more members. The sum is:

[tex]2+4+8+16+...+ 2^{40} [/tex]

To calculate number of ancestors we can use formula for the sum of a geometric sequence. Geometric sequence is sequence of numbers that differ by a certain factor. This factor is called ratio. Formula is:

[tex]S= a_{1} * \frac{1- r^{n} }{1-r} [/tex]

Where:

S -> sum

a1 -> first member of a sequence

r -> ratio

n -> number of elements

For this question:

a1 = 2

r = 2

n = 40

[tex]S= 2* \frac{1- 2^{40} }{1-2} =2 199 023 255 550[/tex]

b.

1 generation = 25 years

40 generations = 40 * 25 = 1000 years

c.

Total number of people who have ever lived = [tex]10^{10} [/tex]

Number of ancestors in 40 generations = [tex]2.2*10^{12} [/tex]

The number of ancestors is greater than total number of people who have ever lived. This means that not all ancestors were distinct and that in each generation both men and women had children with more than one partner.

In the first generation we have 2 ancestors.

In the second generation we have 4 ancestors or [tex]2^{2} [/tex] ancestors.

In the third generation we have 8 ancestors ot [tex]2^{3} [/tex] ancestors.

We can see that each succesive generation has two times more members. The sum is:

[tex]2+4+8+16+...+ 2^{40} [/tex]

To calculate number of ancestors we can use formula for the sum of a geometric sequence. Geometric sequence is sequence of numbers that differ by a certain factor. This factor is called ratio. Formula is:

[tex]S= a_{1} * \frac{1- r^{n} }{1-r} [/tex]

Where:

S -> sum

a1 -> first member of a sequence

r -> ratio

n -> number of elements

For this question:

a1 = 2

r = 2

n = 40

[tex]S= 2* \frac{1- 2^{40} }{1-2} =2 199 023 255 550[/tex]

b.

1 generation = 25 years

40 generations = 40 * 25 = 1000 years

c.

Total number of people who have ever lived = [tex]10^{10} [/tex]

Number of ancestors in 40 generations = [tex]2.2*10^{12} [/tex]

The number of ancestors is greater than total number of people who have ever lived. This means that not all ancestors were distinct and that in each generation both men and women had children with more than one partner.