(2) questions on convergent and divergent

So, 1/n^2 < 1/(n-1)(n+4), and thus the series is less than a convergent series.In summary, for the first conversation, after 10 years the spill will have spread 7260 meters from its source if the pattern continues. It is possible for the spill to reach the grounds of a school located 4000 meters away from the source, depending on the continued pattern of spreading.For the second conversation, the first series is convergent and the second series is divergent. The first series can be solved by finding the sum of the first n terms of a particular series, while the second series can be determined to be convergent by comparing it to a similar series.
  • #1
whitehorsey
192
0
1. A year after the leak began the chemical had spread 1500 meters from its source. After two years, the chemical had spread 900 meters more, and by the end of the third year, it had reached an additional 540 meters.
a. If this pattern continues, how far will the spill have spread from its source after 10 years?
b. Will the spill ever reach the grounds of a school located 4000 meters away from the source explain? Explain.


2. None


3. I tried but i can't seem to find the pattern for this problem.
1500, 2400, 2940...


2nd question​

1. Use the comparison test to determine whether each series is convergent or divergent.
1/2 + 1/9 + 1/28 + 1/65 + ...


2. None


3. I tried to solve this problem but I can't seem to find the pattern for it Can you guys show me? Thank You!.
 
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  • #2
Instead of the total distance, try to first see a connection in the additional distance reached every year.
1500, 900, 540, ...

You will find this is a particular kind of series, and the total distance it has spread from the source in year n is the sum of the first n terms of this series.

For the second one, all you need is something to compare it to. Maybe it helps to note that, 9 <= 9, 16 < 28, 25 < 65, etc.
 
  • #3


1. I would suggest using mathematical models and equations to analyze the data and predict the future spread of the chemical. Based on the given information, the pattern seems to be decreasing by a factor of 0.6 after each year. Therefore, after 10 years, the spill would have spread approximately 3,582 meters from its source.

b. It is possible that the spill may reach the grounds of a school located 4000 meters away from the source, depending on the rate of spread and any potential interventions or containment measures taken. However, it is also important to consider other factors such as wind patterns and topography that may affect the spread of the chemical.

2. I would recommend using the comparison test to determine the convergence or divergence of the series. The comparison test states that if the terms of a series are always smaller than the terms of a convergent series, then the original series must also be convergent. In this case, the given series can be compared to the convergent series 1/n^2. Since 1/n^2 is always larger than the terms in the given series, it can be concluded that the given series is also convergent.

3. For the first problem, the pattern is not as straightforward as it may seem. It may require more data or information to accurately predict the future spread of the chemical. As for the second problem, the pattern is not as obvious and may require some algebraic manipulation to find a common pattern or ratio between the terms. It is important to use mathematical methods and tools to accurately solve these problems.
 

1. What is the difference between convergent and divergent thinking?

Convergent thinking involves narrowing down options to find the single best solution to a problem, while divergent thinking involves generating multiple ideas and possibilities to solve a problem. Convergent thinking is more focused and analytical, while divergent thinking is more open and creative.

2. How can I improve my convergent and divergent thinking skills?

To improve convergent thinking, you can practice problem-solving and decision-making exercises. To improve divergent thinking, you can engage in activities that encourage brainstorming and creative thinking, such as mind mapping or free writing.

3. Are convergent and divergent thinking equally important in the scientific process?

Yes, both types of thinking are important in the scientific process. Convergent thinking helps scientists narrow down and evaluate potential solutions to a problem, while divergent thinking allows them to generate new ideas and hypotheses to explore.

4. Are there any real-life examples of convergent and divergent thinking in science?

Yes, there are many examples of both types of thinking in science. For instance, a scientist may use convergent thinking to select the most effective treatment for a disease, while using divergent thinking to come up with new research ideas to better understand the disease.

5. Can convergent and divergent thinking be used together?

Absolutely. In fact, the most effective problem-solving often involves a combination of both convergent and divergent thinking. This allows for thorough evaluation of potential solutions while also exploring new and innovative ideas.

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