Chapter 12.2, Problem 46E

To determine

To calculate:

Number of descendants does this ancestor have in fifteenth generation.

Answer to Problem 46E

Number of descendants does this ancestor have in fifteenth generation is $620881$ .

Explanation of Solution

Given information:

Each descendant has an average of $2.5$ children.

Calculation:

The number of descendants of each generation forms a geometric sequence with first term $1$ and common ratio $2.5$ and it has $15$ terms.

The number of descendants can be written as,

  $1,2.5,{\left(2.5\right)}^2,{\left(2.5\right)}^3$

Sum of first $n$ terms of the geometric sequence having first term $a$ and common ratio $r$ is,

  $S_n=\frac{a\left(r^n-1\right)}{r-1}\mathrm{...}\left(3\right)$

Plug in $n=15,a=1,r=2.5$ in $1$ to get total number of descendants.

  $\begin{array}{l}{S}_{15}=\frac{1{\left(2.5\right)}^{15}-1}{2.5-1}\\ S_{15}=620881\end{array}$

Thus, Number of descendants does this ancestor have in fifteenth generation is $620881$ .