Let's break this problem down step by step.
Part (a): Showing it forms a geometric sequence
To show that the number of people with the flu forms a geometric sequence, we need to demonstrate that the ratio between consecutive terms in the sequence is constant.
We are given:
- Day 1: 2 people
- Day 2: 10 people
- Day 3: 50 people
Calculate the ratio of the number of people between consecutive days:
[tex]\[
\text{Ratio from Day 1 to Day 2} = \frac{\text{Number of people on Day 2}}{\text{Number of people on Day 1}} = \frac{10}{2} = 5
\][/tex]
[tex]\[
\text{Ratio from Day 2 to Day 3} = \frac{\text{Number of people on Day 3}}{\text{Number of people on Day 2}} = \frac{50}{10} = 5
\][/tex]
Since the ratio is the same (5) across consecutive days, the number of people with the flu forms a geometric sequence with a common ratio [tex]\( r = 5 \)[/tex].
Part (b): Calculating how many people have the flu after one week (seven days)
In a geometric sequence, the number of people on the [tex]\( n \)[/tex]-th day can be found using the formula:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where:
- [tex]\( a_n \)[/tex] is the number of people on the [tex]\( n \)[/tex]-th day
- [tex]\( a_1 \)[/tex] is the number of people on the first day (initial term)
- [tex]\( r \)[/tex] is the common ratio
- [tex]\( n \)[/tex] is the day number
For Day 7:
[tex]\[
a_7 = 2 \cdot 5^{(7-1)} = 2 \cdot 5^6 = 2 \cdot 15625 = 31250
\][/tex]
Therefore, 31,250 people will have the flu after one week (seven days).
Part (c): Calculating how many people will have the flu after one year
There are 365 days in a non-leap year. We will use the same formula to find the number of people on Day 365:
[tex]\[
a_{365} = 2 \cdot 5^{(365-1)} = 2 \cdot 5^{364}
\][/tex]
Given the enormous power, [tex]\( 5^{364} \)[/tex], calculating this value directly would give an astronomically large number, beyond the realistic capacity in any given population.
Comment on the answer:
The application of the geometric sequence model in this context demonstrates theoretically exponential growth of the flu epidemic, assuming no interventions or limiting factors. However, in a real-world scenario, several factors such as medical interventions, natural immunity, population saturation, and other elements would prevent such indefinite exponential growth. Therefore, while the mathematical model predicts an extensive spread, realistic constraints would significantly alter this outcome.
