Question

Before email made it easy to contact many people quickly, groups used telephone trees to pass news to their members. In one group, each person is in charge of calling 3 people. One person starts the tree by calling 3 people. At the second stage, each of these 3 people calls 3 new people. In the third stage, each of the people in stage two calls 3 new people, and so on. (a) How many people have the news by the end of the \(5^{\text {th }}\) stage? (b) Write a formula for the total number of members in a tree of 10 stages. (c) How many stages are required to cover a group with 1000 members?

Short answer

Answer: 121 Question: What is the formula for the total number of members in a tree of 10 stages? Answer: (1 - 3^10) / -2 Question: How many stages are required to cover a group with 1000 members? Answer: 7

Step-by-step solution

Part (a): Number of people after the 5th stage

To find the number of people who have the news by the end of the 5th stage, we'll first list the number of people who receive the news at each stage: - Stage 1: 1 (initial person, who already has the news) - Stage 2: 3 (1 person from the first stage calls 3 new people) - Stage 3: 3*3 = 9 (each of the 3 people from the second stage calls 3 new people) - Stage 4: 3*3*3 = 27 (each of the 9 people from the third stage calls 3 new people) - Stage 5: 3*3*3*3 = 81 (each of the 27 people from the fourth stage calls 3 new people) Now, we'll add these numbers to find the total people who have the news after the 5th stage. Total = 1 + 3 + 9 + 27 + 81 = 121 So, the total number of people who have the news by the end of the 5th stage is 121.

Part (b): Formula for the total number of members in a tree of 10 stages

Notice that at the end of each stage, we are basically adding a geometric series with ratio 3. With this observation, we can write the formula for the number of people who have the news after n stages: Total after n stages = 1 + 3 + 3^2 + 3^3 + ... + 3^(n-1) This is a geometric series with the first term a = 1 and common ratio r = 3. The formula to add a geometric series with n terms is given by: S_n = a * (1 - r^n) / (1 - r) For 10 stages, n = 10: S_10 = 1 * (1 - 3^10) / (1 - 3) = (1 - 3^10) / -2 So, the formula for the total number of members in a tree of 10 stages is (1 - 3^10) / -2.

Part (c): Number of stages required to cover a group with 1000 members

We can use the formula for the number of people who have the news after n stages, and find the smallest n for which the total number of members is greater than or equal to 1000. Using the geometric series formula: 1 + 3 + 3^2 + ... + 3^(n-1) >= 1000 (1 - 3^n) / -2 >= 1000 3^n <= 1 - 2 * 1000 Now we'll find the smallest integer n that satisfies the above inequality: 3^0 = 1 (Not enough people) 3^1 = 3 (Not enough people) 3^2 = 9 (Not enough people) ... 3^6 = 729 (Not enough people) 3^7 = 2187 (Covers the group) So, the smallest n that satisfies the inequality is 7, which means it requires 7 stages to cover a group with 1000 members.

Key concepts

Exponential Growth

In many real-world scenarios, events can grow at an exponential rate, meaning they increase rapidly relative to the size of the current number of events. A perfect example is a telephone tree, where each caller reaches multiple new callers. This process accelerates the spread of information dramatically. Initially, it might seem slow — with just a couple of people knowing — but each new stage vastly increases reach.
For instance, let's look at a situation where each person informs three new people. If this process is repeated multiple times, the growth in the number of informed people is exponential. Every stage triples the number of people involved, leading to rapid expansion.
Understanding exponential growth is crucial because it helps in predicting how quickly something like information can spread. This is not just confined to telephone trees but is seen in natural phenomena like population growth or the spread of viruses, where the number of affected individuals increases dramatically in a short period. Recognizing this pattern is key to planning and managing resources effectively in various fields.

Stages of Growth

Each stage in a growth process can have its distinct characteristics, impacting the overall progression. In our telephone tree example, every stage involves each person reaching out to three others, multiplying the outreach by three at every stage. This multiplication affects the total number of informed individuals cumulatively rather than each stage being independent.
Breaking down the stages:

• In the first stage, the initiator informs three people.
• In the second stage, those three inform another three each, resulting in nine new individuals.
• By the third stage, those nine inform three more people each, adding 27 new individuals to the chain.

This method of compounding at each stage shows how quickly communication or growth can escalate over a few iterations. When planning for growth or spreading information, it is beneficial to consider how each successive stage can expand much more significantly than the previous one. Recognizing and planning for these stages can be highly beneficial in projects, communications, and strategy building.

Geometric Sequence

The concept of a geometric sequence plays a significant role in understanding processes that involve repeated multiplication, like our telephone tree scenario. A geometric sequence is a series of numbers where each term after the first is a fixed multiple of the preceding one. Here, the multiplier is the same at each step - known as the common ratio.
In our example, the numbers of people called in each stage form a geometric sequence: 1, 3, 9, 27, and so forth, with a common ratio of 3. Each successive term is obtained by multiplying the previous term by 3. This regularity aids in predicting outcomes further down the sequence; for instance, by using formulas designed for geometric sequences.
The formula for the total members informed at any stage is derived using the geometric series formula, where the sum of terms in the geometric sequence from the first to the nth term gives us the total number of people informed up to that stage. Recognizing and applying geometric sequences allows us to efficiently compute outcomes across many applications, ranging from simple communication spreads to complex financial modeling.