In this set of exercises, you will use sequences and their...

In this set of exercises, you will use sequences and their sums to study real-world problems. Many large corporations have in place an emergency telephone chain in which one employee in each division is designated to be the first called in the case of an emergency. That employee then calls three employees within the division, each of whom in turn calls three employees, and so on. The chain stops once all the employees have been notified of the emergency. (a) Write the first five terms of the sequence that represents the number of people called at each step of the chain. (The first step consists of just the "designated" employee being called.) Is this an arithmetic sequence or a geometric sequence? (b) Use an appropriate formula to answer the following question: How many steps of the chain are needed to notify all the employees of a corporate division with 600 employees? (c) Explain why this method of notification is very efficient.

In this set of exercises, you will use sequences and their sums to study real-world problems.
Many large corporations have in place an emergency telephone chain in which one employee in each division is designated to be the first called in the case of an emergency. That employee then calls three employees within the division, each of whom in turn calls three employees, and so on. The chain stops once all the employees have been notified of the emergency.
(a) Write the first five terms of the sequence that represents the number of people called at each step of the chain. (The first step consists of just the "designated" employee being called.) Is this an arithmetic sequence or a geometric sequence?
(b) Use an appropriate formula to answer the following question: How many steps of the chain are needed to notify all the employees of a corporate division with 600 employees?
(c) Explain why this method of notification is very efficient.

Key Concepts

Sequences

Sequences are ordered lists of numbers where each term is related to its predecessors by a specific rule. In mathematics, sequences provide a framework to describe patterns and repetitions, which can be leveraged to model diverse phenomena in real-world problems, such as population growth, financial interest, or emergency notifications.

Arithmetic Sequences

An arithmetic sequence is a type of sequence in which the difference between consecutive terms is constant. This constant difference means that each term is obtained by adding a fixed number to the previous term, resulting in a linear progression. Although simple and predictable, arithmetic sequences are less effective at modeling situations that involve rapid or multiplicative growth.

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed, non-zero constant known as the common ratio. This multiplicative process leads to exponential growth or decay, making geometric sequences particularly useful when modeling processes where changes compound over time, such as chain reactions, population growth, or the spread of notifications in an emergency network.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. This concept is important when determining the cumulative total of a quantity that grows exponentially. The sum of a geometric series can be calculated using a specific formula, which is beneficial for solving problems like determining the number of steps required to exceed a certain threshold in notification systems or financial models.

Transcript

00:01 This scenario, the war problem, we have a telephone chain where employees that are notified about an accident.

00:09 So you begin with one employee, and that employee calls three other employees to inform of the emergency situation, and those three employees inform three other employees each, and so on, creating a sort of telephone chain.

00:26 And for part a, in this scenario, we are asked to find the steps of these chain, or writing five terms of the sequence that represent this telephone chain, using what we know about geometric and arithmetic series.

00:44 So we know that it begins with one person.

00:46 So the first person is plus one, one plus, and then the next step is three people, that one person calls three other people.

00:53 And then the next step is that each of these three people, people call three other people of their own.

00:59 So this is this third term is equal to three times three.

01:04 And then as you can see from this pattern, we see another, the fourth step of the chain is that the previous nine people, three times three people.

01:13 From the previous step, each called three, three people of their own.

01:17 So we actually have nine times three or three cute.

01:21 And this is the fourth step.

01:22 And for the fifth step, we see that we actually get each of these three cubed number of people called three people of their own.

01:29 So we get three, three cubed times three, which is equal to three to the fourth power.

01:35 And now that we have this chain for a, we can leave this as the answer.

01:42 And for part b, we're asked to use a formula to identify how many steps of the chain is needed to notify all the employees of a division of 600 total employees.

02:03 So 600 total employees.

02:05 And keep in mind that this is a geometric sequence.

02:08 So geometric sequence.

02:09 And we know that because looking at the terms, we have each term is three times the proceeding.

02:20 And since the pattern is multiplication, each term is being multiplied by a certain number, or the multiplier r.

02:27 Multiply r which is three in this case we actually have um that means that we have a geometric series not an arithmetic series so this is for part b we continue with a scenario where the employees consist of 600 people and so we can use the a formula for the sum of a geometric series which is one which is a1 times so the first term times one minus the multiplier r raised to the n power.

03:00 So the total number of terms divided by one minus r, the multiplier...

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