YouTube Explosion

Task

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.

1. 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					...		...

2. What is the total number of views the video has after hour 3? After hour 4? After hour 10? After hour $n$?
3. 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					...		...

4. How many total people have seen the video via Hillary after hour $n$?
5. 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 $n$.
6. Let $H$ represent your answer to part (d). In terms of $H$, how many people viewed the video via Cory and Michelle after hour $n$?
7. 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 $n$.
8. Using this formula, determine the total number of views the video will have after 10 hours.
9. 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.

IM 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).

Solution

1. Hour	1	2	3	4	...	10	...	$n$
   People Receiving Video	3	9	27	81	$\cdots$	310	...	3$n$

2. We recall that everyone who receives the video watches it exactly once. Therefore, the total number of views after hour $n$ is just the sum of the number of people who received the video during each hour. Hence,
   
   after hour 3, $3 + 9 + 27 = 39$ people have watched the video,
   after hour 4, $3 + 9 + 27 + 81 = 120$ people have watched the video,
   after hour 10, $3 + 9 + 27 + \cdots + 3^{10} = 88,572$ people have watched the video, and
   after hour $n$, $3 + 9 + 27 + \cdots + 3^n$ people have watched the video.

3. Hour	1	2	3	4	$\cdots$	10	$\cdots$	$n$
   People Receiving Video	1	3	9	27	$\cdots$	39	$\cdots$	3$n - 1$

4. The total number of people who have viewed the video via Hillary after hour $n$ is the sum of the number of people who received the video via Hillary after each hour; namely, $1 + 3 + 9 + \cdots + 3^{n-1}$.
5. The number of people who have viewed the video not through Hillary is simply the total number of people who have watched the video, $3 + 9 + \cdots + 3^n$, minus the number of people who have watched the video through Hillary, $1 + 3 + \cdots + 3^{n-1}$. Hence, in terms of $n$, the number of people who have watched the video through Michelle and Cory is $$ (3 + 9 + \cdots + 3^n) - (1 + 3 + \cdots + 3^{n-1}) = 3^n - 1. $$
6. By introducing the variable $H = 1 + 3 + \cdots + 3^{n-1}$, we can view the total number of people who have watched the video as just $3H$. This can be seen either algebraically (by multiplying and distributing) or intuitively/pictorially by seeing that the number of people who have watched the video through Cory is exactly the same as the number of people who have watched it through Hillary; and the same goes for viewers via Michelle. Hence the number of people who have watched the video through Michelle and Cory is $$ 3H - H = 2H. $$
7. Since parts (e) and (f) asked for exactly the same quantity in two different forms, the two expressions we obtained are equal: $$ 2H = 3^n - 1. $$ To determine the value of $H$, we can solve the equation: $$ H = \frac{3^n - 1}{2}. $$
8. Using our formula above, we know that the number of people who have viewed the video via Hillary after hour 10 is $$ \frac{3^{10} - 1}{2} = 29524. $$ Hence the total number of people who have watched the video after hour 10 is $$ 3(29524) = 88572. $$
9. From the context, it is assumed that $r$ is a positive integer. If we are starting off with $r$ friends, and each one is sharing the video with $r$ people, then the total number of views after hour $n$ is $$ r + r^2 + r^3 + \ldots + r^n, $$ where $r$ is the number of people who received the video in hour 1, $r^2$ the number of people who received it in hour 2, and so on. By once again focusing on a single friend's sharing of the video, we see that the number of people who have viewed the video through a single friend after $n$ hours is $$ F = 1 + r + r^2 + \ldots + r^{n-1}. $$ Retracing our steps in parts (e) and (f), we see that the number of people who have viewed the video not through this specific friend after $n$ hours is $$ (r - 1)F = (r + r^2 + \ldots + r^n) - (1 + r + \ldots + r^{n-1}) = r^n - 1, \mbox{ so} $$ $$ F = \frac{r^n - 1}{r - 1}. $$ Thus the total number of views after $n$ hours is $$ rF = \frac{r^{n+1} - r}{r - 1}. $$