Identify The Correct Explicit Formula For The Following Sequence:$\[-15, -20, -25, -30, -35, \ldots\\]A. $a_n = 5n - 10$ B. $a_n = -5n + 15$ C. $a_n = -5n - 15$ D. $a_n = -5n - 10$

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Introduction

In mathematics, sequences are an essential concept used to describe a series of numbers that follow a specific pattern. Identifying the correct explicit formula for a given sequence is a crucial skill that can be applied in various mathematical and real-world problems. In this article, we will explore how to identify the correct explicit formula for the sequence: ${-15, -20, -25, -30, -35, \ldots}$

Understanding the Sequence

The given sequence is a series of negative integers that decrease by a constant difference. To identify the correct explicit formula, we need to analyze the pattern of the sequence and determine the relationship between the terms.

Analyzing the Pattern

Let's examine the given sequence:

{-15, -20, -25, -30, -35, \ldots\}

We can see that each term is decreasing by 5. This suggests that the sequence is an arithmetic sequence with a common difference of -5.

Identifying the Explicit Formula

An explicit formula for a sequence is a mathematical expression that describes the nth term of the sequence. In this case, we need to find the explicit formula that represents the given sequence.

Let's analyze the options:

A. an=5n−10a_n = 5n - 10 B. an=−5n+15a_n = -5n + 15 C. an=−5n−15a_n = -5n - 15 D. an=−5n−10a_n = -5n - 10

To determine the correct explicit formula, we need to substitute the first term of the sequence (-15) into each option and see which one satisfies the equation.

Substituting the First Term

Let's substitute -15 into each option:

A. an=5n−10a_n = 5n - 10 −15=5(1)−10-15 = 5(1) - 10 −15=5−10-15 = 5 - 10 −15=−5-15 = -5

This option does not satisfy the equation.

B. an=−5n+15a_n = -5n + 15 −15=−5(1)+15-15 = -5(1) + 15 −15=−5+15-15 = -5 + 15 −15=10-15 = 10

This option does not satisfy the equation.

C. an=−5n−15a_n = -5n - 15 −15=−5(1)−15-15 = -5(1) - 15 −15=−5−15-15 = -5 - 15 −15=−20-15 = -20

This option does not satisfy the equation.

D. an=−5n−10a_n = -5n - 10 −15=−5(1)−10-15 = -5(1) - 10 −15=−5−10-15 = -5 - 10 −15=−15-15 = -15

This option satisfies the equation.

Conclusion

Based on our analysis, the correct explicit formula for the sequence ${-15, -20, -25, -30, -35, \ldots}$ is:

an=−5n−10a_n = -5n - 10

This formula represents the nth term of the sequence and satisfies the equation when the first term (-15) is substituted into the formula.

Real-World Applications

Identifying the correct explicit formula for a sequence has numerous real-world applications, including:

  • Finance: Understanding the pattern of a sequence can help investors make informed decisions about investments and financial planning.
  • Science: Identifying the explicit formula for a sequence can help scientists model and analyze complex phenomena, such as population growth and disease spread.
  • Engineering: Understanding the pattern of a sequence can help engineers design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

In conclusion, identifying the correct explicit formula for a sequence is a crucial skill that can be applied in various mathematical and real-world problems. By analyzing the pattern of the sequence and substituting the first term into each option, we can determine the correct explicit formula. This skill has numerous real-world applications and can help individuals make informed decisions in various fields.

References

  • [1] "Sequences and Series" by Khan Academy
  • [2] "Arithmetic Sequences and Series" by Math Open Reference
  • [3] "Explicit Formulas for Sequences" by Wolfram MathWorld

Additional Resources

  • [1] "Sequences and Series" by MIT OpenCourseWare
  • [2] "Arithmetic Sequences and Series" by Purplemath
  • [3] "Explicit Formulas for Sequences" by Mathway
    Q&A: Identifying the Correct Explicit Formula for a Sequence ===========================================================

Introduction

In our previous article, we explored how to identify the correct explicit formula for a sequence. In this article, we will answer some frequently asked questions about identifying the correct explicit formula for a sequence.

Q: What is an explicit formula for a sequence?

A: An explicit formula for a sequence is a mathematical expression that describes the nth term of the sequence. It is a way to express the sequence in a compact and concise form.

Q: How do I identify the correct explicit formula for a sequence?

A: To identify the correct explicit formula for a sequence, you need to analyze the pattern of the sequence and determine the relationship between the terms. You can use the following steps:

  1. Examine the sequence and identify the common difference.
  2. Determine the first term of the sequence.
  3. Substitute the first term into each option and see which one satisfies the equation.

Q: What is the common difference in a sequence?

A: The common difference in a sequence is the constant difference between each term. For example, in the sequence ${-15, -20, -25, -30, -35, \ldots}$, the common difference is -5.

Q: How do I determine the first term of a sequence?

A: The first term of a sequence is the first number in the sequence. For example, in the sequence ${-15, -20, -25, -30, -35, \ldots}$, the first term is -15.

Q: What is the difference between an explicit formula and an implicit formula?

A: An explicit formula for a sequence is a mathematical expression that describes the nth term of the sequence. An implicit formula for a sequence is a mathematical expression that describes the relationship between the terms of the sequence, but does not explicitly state the nth term.

Q: Can I use a calculator to find the explicit formula for a sequence?

A: Yes, you can use a calculator to find the explicit formula for a sequence. However, it is often more efficient and effective to use algebraic methods to find the explicit formula.

Q: What are some real-world applications of identifying the correct explicit formula for a sequence?

A: Identifying the correct explicit formula for a sequence has numerous real-world applications, including:

  • Finance: Understanding the pattern of a sequence can help investors make informed decisions about investments and financial planning.
  • Science: Identifying the explicit formula for a sequence can help scientists model and analyze complex phenomena, such as population growth and disease spread.
  • Engineering: Understanding the pattern of a sequence can help engineers design and optimize systems, such as electrical circuits and mechanical systems.

Q: How can I practice identifying the correct explicit formula for a sequence?

A: You can practice identifying the correct explicit formula for a sequence by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice identifying the correct explicit formula for a sequence.

Conclusion

In conclusion, identifying the correct explicit formula for a sequence is a crucial skill that can be applied in various mathematical and real-world problems. By analyzing the pattern of the sequence and substituting the first term into each option, we can determine the correct explicit formula. We hope that this Q&A article has provided you with a better understanding of how to identify the correct explicit formula for a sequence.

References

  • [1] "Sequences and Series" by Khan Academy
  • [2] "Arithmetic Sequences and Series" by Math Open Reference
  • [3] "Explicit Formulas for Sequences" by Wolfram MathWorld

Additional Resources

  • [1] "Sequences and Series" by MIT OpenCourseWare
  • [2] "Arithmetic Sequences and Series" by Purplemath
  • [3] "Explicit Formulas for Sequences" by Mathway