Question

Some people believe they can make money from a chain letter (they are usually disappointed). A chain letter works roughly like this: A letter arrives with a list of five names attached and instructions to mail a copy to five friends and to send \(\$ 1\) to the top name on the list. When you mail the five letters, you remove the top name (to whom the money was sent) and add your own name to the bottom of the list. (a) If no one breaks the chain, how much money do you receive? (b) Assuming no one broke the chain, how many total letters were written with your name on them?

Short answer

Answer: If no one breaks the chain in the chain letter process, a person would receive a total of $30. There will be 155 letters written with their name on them.

Step-by-step solution

(a) Calculating the money received

To calculate the money received, consider the following steps in the chain: 1. When you add your name to the list and send the letters, the next five people will send \(\$1\) each to the top name (which is now the second name on the list since you removed the previous top name). Since no one breaks the chain, these five people will send you a total of \(5\times \$1=\$5\). 2. Each of these five people now has your name second on their list and will send the chain to five of their friends. This means a total of \(5\times 5=25\) people now have your name on top of their list, and they will in turn send you \(\$1\) each, making a total of \(25\times \$1=\$25\). So, by adding these amounts, you receive \(\$5+\$25=\$30\) in total if no one breaks the chain.

(b) Calculating the total letters with your name on them

Now let's calculate how many total letters will be written with your name on them, assuming no one broke the chain: 1. You send a letter to five friends, so initially there are 5 letters with your name on them. 2. These 5 friends share the letter with 5 of their friends, so they will send out a total of \(5\times 5=25\) letters. Your name is now the second on the list, so there are now \(5+25=30\) letters with your name on them. 3. Each of the initial 25 friends sends the letter to 5 more friends, making a total of \(25\times 5=125\) new letters. Your name is now at the top, therefore, there are \(30+125=155\) letters with your name on them. So, when you are the top name on the list, a total of 155 letters will be written with your name on them.

Key concepts

Money Calculation in Chain Letters

Chain letters have an intriguing method where many people believe they can reap financial benefits simply by forwarding copies of a letter and adding their name to a list. Here's how the money calculation works:

• You start by sending the chain letter to five friends, and you send \\(1 to the person whose name is at the top of the list. You then put your name at the bottom of the list.
• When these five friends receive the letter, if they each follow through by sending out five letters themselves, a total of five people will send \\)1 to you, because you become the second name on their lists. This earns you \\(5.
• As the chain continues, those 25 people (each of your five friends sending the letter to five others) now have you at the top of their lists. Each will send you another \\)1, totaling \\(25.

Therefore, assuming nobody breaks the chain, the total amount of money you could potentially receive is \\)30, calculated by adding \\(5 and \\)25 together.
This simple calculation highlights the mathematical nature of chain letters, although in reality, the chances of everyone participating without breaking the sequence are slim.

Mathematics in Real-Life Scenarios

While chain letters might seem like just a whimsical idea, they provide a fascinating example of how mathematics applies to real-life situations. The whole process exploits basic principles of probability and exponential growth.

• The allure of chain letters lies in their simplicity and the potential for quick financial gain, which is enticing to many.
• From a mathematical standpoint, this scenario demonstrates geometric progression, where each person forwards the letter to multiple new people, exponentially increasing the reach.
• The expectation that everyone in the chain will participate perfectly reflects both the excitement and the improbability of success.

These real-life applications of mathematical concepts serve as a reminder of how underlying mathematical principles can sometimes present us with impractical or even misleading expectations.

Sequence and Series Problems

Chain letters essentially rely on the principles of sequences and series. When you engage in a chain letter, you're entering a sequence of actions that can be analyzed using these mathematical concepts:

• At each step, the number of letters—and by extension, the number of dollars you might receive—is determined by a geometric sequence. Each participant sends out five letters, creating a sequence of multiplication by five.
• The problem of calculating total letters and potential earnings involves finding the sum of a series. This series is geometric, defined by its common ratio, which is consistent with the number of people each participant passes the letter onto.
• For example, if you consider the series for total letters, you start with 5, then 25, then 125, and so on, representing the exponential growth of letters with your name.

Thus, chain letters not only illustrate the excitement of mathematical potential but also serve as a cautionary note on the limitations and risks inherent in assuming infinite participation. Understanding sequences and series gives us insight into the theoretical underpinnings of such schemes.