Chapter 11.5, Problem 12P

To determine

To calculate:

Number of rounds of calls is needed before everyone is contacted.

Answer to Problem 12P

Number of rounds $\left(n\right)$ of calls are needed before everyone is contacted is $6$ .

Explanation of Solution

Given information:

Total number of members: $1560$

Calculation:

Here need to find out the number of elements of the geometric series need to be added to get $1560$ .

  $\begin{array}{l}2{\displaystyle {\sum }_{k=0}^n{3}^k}\\ =2\frac{3^{n+1}-1}{3-1}\\ =3^{n+1}-1=1560\end{array}$

Equation is,

  $3^{n+1}-1=1560$

Take the logarithm on both sides, $n$ is a whole number, so round up $5.7$ ,

  $n+1={\log }_{3}1561\approx 6.7$

  $n=6$

Therefore, Number of rounds $\left(n\right)$ of calls are needed before everyone is contacted is $6$ .