Where does one term end and the other begin?

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In summary, the conversation discusses the concept of where one term ends and the other begins in scientific terminology. It is acknowledged that this boundary can vary depending on context and definitions, and can be determined by consulting authoritative sources or considering the relationship between terms. There are no universal rules for determining term boundaries, and they can change over time due to evolving scientific knowledge and technologies. It is important to clearly define term boundaries for effective communication, but also recognize that they can be subject to change and interpretation.
  • #1
trigger352
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I need help determining the explicit formula and writing the series in sigma notation with a specified lower limit.

In both of them, I cannot tell where one term ends and the other begins, or what to do when the sign changes.

Series #1
[tex]- 2 - 5 - 8 - 11 - 14 - 17 - 20[/tex]; lower limit = [tex]3[/tex]
Series #2
[tex]4.4 + 2.4 + 0.4 - 1.6 - 3.6 - 5.6 - 7.6[/tex]; lower limit = [tex]2[/tex]
 
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  • #2
Each term is a number, either positive or negative. Since this is a series, you are implicitly adding all the terms. The terms which are negative get rid of the plus sign before them, per convention.

- Warren
 
  • #3



The explicit formula for Series #1 is given by a_n = -3n + 1, where n is the term number. To write this series in sigma notation with a lower limit of 3, we can use the following expression:

∑(n=3 to ∞) (-3n + 1)

Similarly, the explicit formula for Series #2 is a_n = 6 - 2n, where n is the term number. To write this series in sigma notation with a lower limit of 2, we can use the following expression:

∑(n=2 to ∞) (6 - 2n)

In terms of where one term ends and the other begins, it is important to note that in both series, the terms are being subtracted from each other. So in Series #1, the first term is -2, the second term is -5, and the difference between these two terms is -3. This pattern continues for the rest of the series. Similarly, in Series #2, the first term is 4.4, the second term is 2.4, and the difference between these two terms is -2. This pattern also continues for the rest of the series.

Therefore, in both series, the sign change indicates the end of one term and the beginning of the next. To find the sum of these series, we can use the formula for a finite arithmetic series: S_n = (n/2)(a_1 + a_n), where n is the number of terms and a_1 and a_n are the first and last terms, respectively.
 

Related to Where does one term end and the other begin?

1. Where does one term end and the other begin?

This question is often asked in the context of scientific terminology, particularly in the fields of biology and chemistry. It refers to the separation between two distinct terms or concepts. The answer to this question is that it depends on the specific context and the definitions of the terms in question. In some cases, the boundary between terms may be clearly defined, while in others it may be more subjective and open to interpretation.

2. How can one determine where one term ends and the other begins?

There are several ways to determine the boundary between two terms. One approach is to consult authoritative sources, such as dictionaries or scientific textbooks, which provide clear definitions and distinctions between terms. Another approach is to consider the context in which the terms are being used and their relationship to other concepts. Additionally, understanding the etymology or origins of the terms can also provide insight into their boundaries.

3. Are there any universal rules for determining where one term ends and the other begins?

No, there are no universal rules for determining the boundaries between terms. The definition and separation of terms can vary depending on the field of study, the context, and the perspective of the individual using the terms. What may be considered one term in one context may be divided into multiple terms in another context, highlighting the subjective nature of term boundaries.

4. Can the boundary between terms change over time?

Yes, the boundary between terms can change over time. As scientific knowledge and understanding evolve, new terms may be introduced, and existing terms may be redefined or divided into multiple terms. Additionally, the boundaries between terms may also shift as new technologies and methodologies are developed, providing new ways to categorize and understand scientific concepts.

5. How important is it to clearly define the boundaries between terms in science?

Defining the boundaries between terms in science is crucial for effective communication and understanding. Without clear definitions, there can be confusion and misunderstandings, hindering scientific progress. However, it is also important to recognize that term boundaries are not always rigid and can be subject to change and interpretation. Therefore, it is essential to continuously evaluate and refine the definitions of terms in science to ensure accuracy and clarity.

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