Problem 32 - Algebra 2

Question

Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose the first person in the chain calls four people. Then each of these people calls four others, and so on. A Make a tree diagram to show the first three stages in the telephone chain. How many calls are made at each stage? b. Write the series that represents the total number of calls made through the first six stages. c. How many employees have been notified after stage six?

Step-by-Step Solution

Answer

The number of calls at each stage: stage 1 - 1 call (to 4 people), stage 2 - 4 new calls (to 16 people), stage 3 - 16 new calls (to 64 people). The series representing the number of calls through the first six stages: \(1 + 4 + 4^2 + 4^3 + 4^4 + 4^5\). The number of employees notified after six stages is \(\frac{{1 * (4^6 - 1)}}{{4 - 1}}\), which equals to 1364.

Step 1: Draw a Tree Diagram

The first node will represent the initial person who starts the chain who has four connections. At stage two, each of these four people will also call four more people, resulting in \(4 * 4 = 16\) new calls. The third stage sees each of these 16 individuals calling 4 new people each, for a total of \(16 * 4 = 64\) calls.

Step 2: Write the Series

This is a geometric series where the common ratio is 4 (each person calls four others). So, the series of people contacted up to n stages can be written in the form \(1, 4, 4^2, 4^3, ..., 4^{n-1}\). From this, the series for the total number of calls for the first six stages would be \(1 + 4 + 4^2 + 4^3 + 4^4 + 4^5\).

Step 3: Calculate Total Employees Notified

Using the formula for the sum of a geometric series: \[S_n = \frac{{a * (r^n - 1)}}{{r - 1}}\] where \(S_n\) is the sum of the first n terms, a is the first term (1 in this case), r is the common ratio (4), and n is the number of terms (6). So the total employees notified after six stages is \[\frac{{1 * (4^6 - 1)}}{{4 - 1}}\]

Key Concepts

Tree Diagram

A tree diagram is a visual representation that helps to illustrate relationships in a structured, hierarchical way. Each 'stage' or 'level' in the tree diagram represents one round of 'calls' made in the scenario.

The tree starts with a single 'root' node, the first person making calls, who branches out to call four others. These four recipients form the second layer of the tree. Then, in the next stage, each of these four individuals also calls four more individuals.

As you move to the third level, the tree branches further, with each person calling four more, resulting in 16 calls. This pattern continues, significantly multiplying the number of individuals notified at each stage.

• Stage 1: One person calls four others (4 calls)
• Stage 2: Each of these four calls four more (16 calls)
• Stage 3: Each of the 16 calls four more (64 calls)

This diagrammatic approach simplifies understanding how the calls spread, making it easier to visualize large, complex connections.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. In the telephone chain problem, this sequence begins with one caller and uses the common ratio of 4, because each caller notifies four more people.

The progression for the given problem is:

• First person: 1 call
• First stage: 4 calls
• Second stage: 4^2 = 16 calls
• Third stage: 4^3 = 64 calls

The pattern continues, showcasing how rapidly the count of calls escalates with each successive stage. It's crucial to understand that this multiplication creates a rapid expansion, typical of geometric progressions, which differs fundamentally from arithmetic progressions where quantities increase linearly.

Sum of a Geometric Series

To find out how many employees have been notified after a certain number of stages in the chain, we need to calculate the sum of a geometric series. This series is represented by the sum of the first few terms in the geometric progression, which includes all calls made.

In our scenario, the terms are the number of calls at each stage: \(1, 4, 4^2, 4^3, 4^4, 4^5\).
The total number of employees notified across these stages can be found using the sum formula for a geometric series, \(S_n = \frac{a(r^n - 1)}{r - 1}\). Here:

• \(a = 1\)
• \(r = 4\)
• \(n = 6\)

So the total is \(S_6 = \frac{1(4^6 - 1)}{4 - 1} = \frac{4096 - 1}{3} = 1365\).
This calculation shows that 1365 employees have been notified after six stages, exponentially increasing as each person informs four others, demonstrating the impressive power of geometric growth.