Michelle, Hillary, and Cory created a YouTube video, and have a plan to get as many people to watch it as possible. They will each share the video with 3 of their best friends, and create a caption on the video that says, "Please share this video with 3 of your best friends." Each time the video is shared with someone, that person instantly views the video only once and sends it to exactly 3 more people. In addition, assume that every person only receives the video once.
Suppose that after 1 hour since the video was posted, Michelle, Hillary, and Cory have watched the video; after 2 hours, all of their friends have watched the video; and so on.
- Fill in the table for the number of people who received the video during the given hour. (Assume that Michelle, Hillary, and Cory "received" the video as well.)
Hour | 1 | 2 | 3 | 4 | ... | 10 | ... | \(n\) |
People Receiving Video | | | | | ... | | ... | |
- What is the total number of views the video has after hour 3? After hour 4? After hour 10? After hour \(n\)?
- Let's look at only the people who have viewed the video via Hillary. Fill in the following table for the number of people who have received the video via Hillary in the given hour.
Hour | 1 | 2 | 3 | 4 | ... | 10 | ... | \(n\) |
People Receiving Video | | | | | ... | | ... | |
- How many total people have seen the video via Hillary after hour \(n\)?
- Using the generalized (hour \(n\)) expressions you developed in parts (b) and (d), give an expression, in terms of \(n\), for the number of people who viewed the video not via Hillary after hour \(n\).
- Let \(H\) represent your answer to part (d). In terms of \(H\), how many people viewed the video via Cory and Michelle after hour \(n\)?
- Using your answers in questions (e) and (f), write a formula (not a sum) in terms of \(n\) for the number of people who have viewed the video via Hillary after hour \(n\).
- Using this formula, determine the total number of views the video will have after 10 hours.
- Suppose that instead of 3 friends creating a video and sending it to their 3 best friends, we started off with \(r\) friends creating a video and each sending it to \(r\) of their best friends. Write a formula (not a sum) for the number of views the video will have after \(n\) hours.
Commentary
This task, along with associated commentary and solutions, was authored by Lacey Dixon and Jonathan Utegaard of The University of Arizona.
The purpose of this task is to have students derive the formula for the sum of a specific finite geometric series. In determining the total number of views the YouTube video has, students will first come across the sum of the terms of a geometric sequence. Without a formula, students will have to calculate this sum by adding each term individually. By the end of the task, the students will have come up with a formula that will help them find the sum much quicker than by rote calculation.
Within the classroom, this task is made to be used as a group work activity with moments where the whole class comes together to share solutions and thought processes. Teachers can utilize student work, figures, drawings, equations, and interesting thinking that arise throughout this task. The critical piece to the task is recognizing that parts (e) and (f) ask the same thing, but call for two different representations in order to derive a formula in part (g). Thus teachers should keep an eye out for these two different representations and suggest that students share their work with other groups.
One possible concern when having students complete the task is that students might not interpret the "views via Hillary" as intended. The phrase "via Hillary" is meant to signify all of the people who have watched the video because of Hillary. This includes not only the three people to whom Hillary sent the video, but also the people those three sent it to, and so on.
Note that this task does not derive the generalized formula for the sum of every finite geometric series, although having students discover that formula could be a natural extension of this task. The last question in the task would be a good starting point for the general derivation, and the teacher can provide intuitive examples of the meaning of the formula when \(r\) is something other than a whole number (e.g., area representations when \(r\) is a fraction).