For the first 30 days of a flu outbreak, the number of students on your campus who become ill is increasing. Which is worse: The number of students with the flu is increasing arithmetically or is increasing geometrically?
could be either, depending on the rate.
Increasing by 5 per day might be a lot worse than a 0.01% increase
Or, a 2% increase might get a lot worse than 25 per day.
Sounds like a sequence problem since the question is asking for arithmetic or geometric increases. There are formulas associated with these types of sequences but let's look at the BASIC textbook definitions:
In arithmetic sequences each term after the first differs from the one before it by a constant number. Also known as common difference. The common difference doesn't always have to be the same in the sequence but can vary. However, there is usually a pattern. For example, in the sequence:
2, 3, 5, 9, 17 we see that you add each term after 2 by 1, 2, 4, 8. So 2 + 1 = 3, 3 + 2 = 5, etc. However, in the sequence: 3, 5, 7, 9 you simply add each term after 3 by 2.
In geometric sequences each term after the first is multiplied by a non-zero constant number. Otherwise known as the common ratio.
For example: 1 , 2 , 4 , 8 , 16 each successive term is multiplied by 2 which is the common ratio. the common ratio(s): r = 2/1 = 2, r = 4/2 = 2, 8/4 = 2, etc.
In short, when something increases geometrically it grows exponentially and arithmetically usually means linearly. Diseases like the flu or meningitis in a high population environment like a college campus will most likely spread or geometrically. But you can make your own determination.
In error, I told you the common difference varies. That is incorrect. Hence the word, common.
CORRECTION: The sequence
2, 3, 5, 9, 17 is neither geometric nor arithmetic even though we add each term after 2 by 1, 2, 4, 8. So 2 + 1 = 3, 3 + 2 = 5, etc. There is no common ratio either.
Sorry about that.
To determine which scenario is worse, whether the number of students with the flu is increasing arithmetically or geometrically, we need to understand the difference between the two types of progressions.
1. Arithmetic progression (AP): In an arithmetic progression, the subsequent terms are obtained by adding a constant difference to the previous term. For example, if the initial number of students with the flu is 10 and the difference is 5, then the progression would be 10, 15, 20, 25, and so on.
2. Geometric progression (GP): In a geometric progression, the subsequent terms are obtained by multiplying a constant ratio with the previous term. For example, if the initial number of students with the flu is 10 and the ratio is 2, then the progression would be 10, 20, 40, 80, and so on.
Now, let's compare the two scenarios:
If the number of students with the flu is increasing arithmetically, it implies that a constant number of new cases is being added each day. In this case, the pace of the outbreak is relatively steady and predictable.
On the other hand, if the number of students with the flu is increasing geometrically, it means that the number of new cases is multiplying each day, resulting in an exponential growth. This scenario indicates a rapid spread of the flu and could potentially lead to a much larger number of cases.
To determine which is worse, we need more information about the rate of increase in each scenario. However, in general, a geometric progression suggests a faster and potentially more severe outbreak, while an arithmetic progression suggests a slower and more controlled spread.
Therefore, if the number of students with the flu is increasing geometrically, it would generally be considered worse than if it were increasing arithmetically.