Certainly! Let's solve this step-by-step. We need to find the total views of a video over 21 days, with the number of views increasing by 8% each day starting with 500 views on the first day.
1. Start with the initial conditions:
- Initial views on the first day = 500
- Daily increase rate = 8% = 0.08
- Number of days to calculate = 21
2. Define the growth pattern:
The number of views on any given day can be represented by a geometric sequence where each term (number of views on day [tex]\( n \)[/tex]) is:
[tex]\[
V_n = V_1 \times (1 + r)^{n-1}
\][/tex]
where [tex]\( V_1 \)[/tex] is the initial number of views, [tex]\( r \)[/tex] is the daily increase rate, and [tex]\( n \)[/tex] is the day number.
3. Sum the views over 21 days:
The total views over the course of 21 days is the sum of the daily views from day 1 to day 21.
The sum [tex]\( S \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[
S_n = V_1 \times \frac{(1 + r)^n - 1}{r}
\][/tex]
where [tex]\( S_n \)[/tex] is the sum of the first [tex]\( n \)[/tex] terms, [tex]\( r \)[/tex] is the common ratio (the increase rate), and [tex]\( V_1 \)[/tex] is the initial term.
4. Plug in the known values:
- [tex]\( V_1 = 500 \)[/tex]
- [tex]\( r = 0.08 \)[/tex]
- [tex]\( n = 21 \)[/tex]
So the equation for the total number of views [tex]\( S_{21} \)[/tex] over 21 days becomes:
[tex]\[
S_{21} = 500 \times \frac{(1 + 0.08)^{21} - 1}{0.08}
\][/tex]
5. First, compute the growth factor:
[tex]\[
(1 + 0.08)^{21} = 1.08^{21} \approx 4.931
\][/tex]
6. Calculate the sum using the formula:
[tex]\[
S_{21} = 500 \times \frac{4.931 - 1}{0.08}
\][/tex]
[tex]\[
S_{21} = 500 \times \frac{3.931}{0.08}
\][/tex]
[tex]\[
S_{21} = 500 \times 49.1375
\][/tex]
[tex]\[
S_{21} \approx 24568.75
\][/tex]
7. Round the result to the nearest whole number:
[tex]\[
\text{Total views} \approx \text{round}(24568.75) = 24569
\][/tex]
Thus, the total number of views the video got over the course of the first 21 days is approximately 24,569 views.
