Chapter 9.5, Problem 39PPSE

Solution

a.

To make a diagram to show the first three stages in the telephone chain, and tell how many calls are made at each stage.

The numbers of calls made at first three stages are 4, 16, and 64.

Given information:

A company’s CEO calls four people and each of these four people call another four people.

Calculation:

Observe the at first, there are 4 phone calls made, 16 phone calls are made at second stage, 64 phones are made at third stage.

Therefore, the related diagram representing the number of calls made for the first three stage can be drawn as shown below.

  

b.

To find the number of employees that have been notified after stage six.

  $4+16+64+256+1024+4096$

Given information:

A company’s CEO calls four people and each of these four people call another four people.

Calculation:

The calls made through the first three stages are 4, 16 and 64.

This a geometric sequence with first 4 and common ratio 4.

Therefore, the geometric series with six terms will be $4+16+64+256+1024+4096$ .

c.

To write the series that represents the number of calls made through the first six stages.

5460 employees were notified after stage six.

Given information:

A company’s CEO calls four people and each of these four people call another four people.

Calculation:

Find the number of employees that have been notified after the stage by finding the sum of the series using the formula, $S_n=\frac{a_1\left(1-r^n\right)}{1-r}$ .

  $\begin{array}{c}4+16+64+256+1024+4096=\frac{4\left(1-4^6\right)}{1-4}\\ =\frac{4\left(1-4096\right)}{-3}\\ =\frac{4\left(4095\right)}{3}\\ =4\cdot 1365\\ =5460\end{array}$

Therefore, 5460 employees were notified after stage six.