Chapter 12.3, Problem 72E

To determine

To find: The number of ancestors a person has 15 generations ago.

Answer to Problem 72E

The number of ancestors a person has 15 generations ago is 32,768.

Explanation of Solution

Formula used:

The formula to calculate common ratio of a G.P. is,

$\text{Common ratio}\left(r\right)=\frac{a_2}{a_1}$

The $n\text{th}$ term formula for geometric sequence is, $a_n=a{r}^{n-1}$

Given:

A person has two parents, four grandparents, eight great-grandparents and so on.

Calculation:

The sequence formed from the given statement is, $2,4,8,16,\cdots$

Substitute 4 for $a_2$ and 2 for $a_1$ in $\text{Common ratio}\left(r\right)=\frac{a_2}{a_1}$ as follows,

$\begin{array}{c}\text{Common ratio}\left(r\right)=\frac{a_2}{a_1}\\ =\frac{4}{2}\\ =2\end{array}$

Substitute 2 for $a$ and 2 for $r$ and 15 for n in $a_n=a{r}^{n-1}$ to find the number of ancestors,

$\begin{array}{l}{a}_{15}=2{\left(2\right)}^{15-1}\\ a_{15}=2^{15}\\ a_{15}=32768\end{array}$

Therefore, the number of ancestors a person has 15 generations ago is 32,768.