One day you saw someone video on Youtube. At 1pm you shared a video link to 5 unique people. Then at 2pm each of your friends shared it to 5 unique people. Then at 3pm each of their friends shared it with 5 unique people. If this pattern, kept happening, then how many unique people received tis link 11 pm?

To solve this problem, we can analyze the situation by considering how the number of people receiving the link grows each hour. This is a geometric progression where each person shares the video link with 5 unique people every hour.

### Step-by-Step Analysis:

1. **1:00 PM** - You shared the video with 5 people.  
   - Number of unique people at 1 PM = \( 5 \)

2. **2:00 PM** - Each of these 5 people shared the video with 5 more unique people.  
   - Number of unique people at 2 PM = \( 5 \times 5 = 25 \)

3. **3:00 PM** - Each of these 25 people shared the video with 5 more unique people.  
   - Number of unique people at 3 PM = \( 25 \times 5 = 125 \)

4. **4:00 PM** - Each of these 125 people shared the video with 5 more unique people.  
   - Number of unique people at 4 PM = \( 125 \times 5 = 625 \)

5. **Continuing this pattern** - We observe that the number of people receiving the video each hour forms a geometric sequence where:
   - The first term \( a_1 = 5 \)
   - The common ratio \( r = 5 \)

The number of unique people at each subsequent hour is:
\[
a_n = 5^n
\]

### Calculating Total Unique People by 11 PM:

The hours from 1 PM to 11 PM span 10 hours (from 1 to 11 PM).

Let's sum the total number of unique people who received the link over these hours.

- At 1 PM, the total is \( 5 \)
- At 2 PM, the total is \( 5^2 = 25 \)
- At 3 PM, the total is \( 5^3 = 125 \)
- ...
- At 11 PM, the total is \( 5^{11} \)

The total number of unique people who received the link from 1 PM to

Explore the growth of unique recipients of a shared video link through a geometric progression. Understand how each hour's shares expand exponentially, starting with 5 initial shares and a common ratio of 5. Suitable for students in Grades 7-9.

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