Problem Analysis:
The problem describes a scenario where a video link is shared in a geometric progression. Initially, at 1pm, the video is shared with 5 unique people. At 2pm, each of these 5 people shares the video with 5 unique people, resulting in \( 5 \times 5 = 25 \) new people. At 3pm, each of these 25 people shares the video with 5 unique people, resulting in \( 25 \times 5 = 125 \) new people, and so on.
This pattern continues until 11pm, which is 10 hours after 1pm. We need to determine the total number of unique people who received the video link by 11pm.
Step-by-Step Solution:
#### 1.
Identify the Pattern:
- At 1pm: \( 5 \) people receive the video.
- At 2pm: Each of the 5 people shares with 5 new people, so \( 5 \times 5 = 25 \) new people receive the video.
- At 3pm: Each of the 25 people shares with 5 new people, so \( 25 \times 5 = 125 \) new people receive the video.
- This pattern continues, with the number of new people receiving the video at each hour forming a geometric sequence.
#### 2.
Geometric Sequence:
- The number of new people receiving the video at each hour forms a geometric sequence:
\[
5, 25, 125, 625, \ldots
\]
- Here, the first term \( a = 5 \) and the common ratio \( r = 5 \).
#### 3.
Sum of the Geometric Series:
- The total number of unique people who received the video by 11pm is the sum of the first 11 terms of this geometric sequence (from 1pm to 11pm).
- The formula for the sum of the first \( n \) terms of a geometric series is:
\[
S_n = \frac{a(1 - r^n)}{1 - r}
\]
- Here:
- \( a = 5 \) (first term)
- \( r = 5 \) (common ratio)
- \( n = 11 \) (number of terms, from 1pm to 11pm)
#### 4.
Substitute the Values:
- Substitute \( a = 5 \), \( r = 5 \), and \( n = 11 \) into the formula:
\[
S_{11} = \frac{5(1 - 5^{11})}{1 - 5}
\]
- Simplify the denominator:
\[
1 - 5 = -4
\]
- The formula becomes:
\[
S_{11} = \frac{5(1 - 5^{11})}{-4}
\]
- Simplify further:
\[
S_{11} = \frac{5(5^{11} - 1)}{4}
\]
#### 5.
Calculate \( 5^{11} \):
- Compute \( 5^{11} \):
\[
5^{11} = 48828125
\]
#### 6.
Substitute \( 5^{11} \) Back into the Formula:
- Substitute \( 5^{11} = 48828125 \) into the formula:
\[
S_{11} = \frac{5(48828125 - 1)}{4}
\]
- Simplify the expression inside the parentheses:
\[
48828125 - 1 = 48828124
\]
- The formula becomes:
\[
S_{11} = \frac{5 \times 48828124}{4}
\]
- Perform the multiplication:
\[
5 \times 48828124 = 244140620
\]
- Divide by 4:
\[
\frac{244140620}{4} = 61035155
\]
Final Answer:
The total number of unique people who received the video link by 11pm is:
\[
\boxed{61035155}
\]
Parent Tip: Review the logic above to help your child master the concept of arithmetic and geometric sequences word problems worksheet.
