Select The Correct Answer.Consider The First Four Terms Of The Sequence Below: \[$-3, -12, -48, -192, \ldots\$\]What Is The \[$8^{\text{th}}\$\] Term Of This Sequence?A. \[$-768\$\]B. \[$-12,288\$\]C.

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Introduction

In mathematics, a sequence is a series of numbers in a specific order. Identifying the pattern or rule governing a sequence is crucial to determine its subsequent terms. The given sequence is −3,−12,−48,−192,…{-3, -12, -48, -192, \ldots}. In this article, we will analyze the sequence, identify its pattern, and determine the 8th term.

Analyzing the Sequence

To understand the sequence, let's examine the given terms:

  • The first term is −3{-3}.
  • The second term is −12{-12}, which is −3×4{-3 \times 4}.
  • The third term is −48{-48}, which is −12×4{-12 \times 4}.
  • The fourth term is −192{-192}, which is −48×4{-48 \times 4}.

From the above analysis, it is evident that each term is obtained by multiplying the previous term by 4{4}. This indicates that the sequence is a geometric progression with a common ratio of 4{4}.

Determining the 8th Term

Now that we have identified the pattern of the sequence, we can determine the 8th term. To do this, we will start with the first term and multiply it by the common ratio 4{4} seven times, as we need to find the 8th term.

  • The first term is −3{-3}.
  • The second term is −3×4=−12{-3 \times 4 = -12}.
  • The third term is −12×4=−48{-12 \times 4 = -48}.
  • The fourth term is −48×4=−192{-48 \times 4 = -192}.
  • The fifth term is −192×4=−768{-192 \times 4 = -768}.
  • The sixth term is −768×4=−3072{-768 \times 4 = -3072}.
  • The seventh term is −3072×4=−12288{-3072 \times 4 = -12288}.
  • The eighth term is −12288×4=−49152{-12288 \times 4 = -49152}.

Therefore, the 8th term of the sequence is −49152{-49152}.

Conclusion

In conclusion, the sequence −3,−12,−48,−192,…{-3, -12, -48, -192, \ldots} is a geometric progression with a common ratio of 4{4}. By analyzing the pattern of the sequence, we were able to determine the 8th term, which is −49152{-49152}.

Answer

The correct answer is −49152{-49152}.

Discussion

  • What is the common ratio of the sequence?
  • How can we determine the nth term of a geometric progression?
  • What is the difference between an arithmetic progression and a geometric progression?

References

Related Topics

Introduction

In our previous article, we analyzed a geometric progression and determined the 8th term of the sequence. In this article, we will answer some frequently asked questions related to geometric progressions and sequences.

Q: What is a geometric progression?

A: A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the formula for the nth term of a geometric progression?

A: The formula for the nth term of a geometric progression is given by:

an = ar^(n-1)

where an is the nth term, a is the first term, r is the common ratio, and n is the term number.

Q: How can we determine the common ratio of a geometric progression?

A: To determine the common ratio of a geometric progression, we can divide any term by its previous term. For example, if we have the terms 2, 6, 18, 54, ..., we can divide the second term by the first term to get the common ratio:

r = 6/2 = 3

Q: What is the difference between an arithmetic progression and a geometric progression?

A: An arithmetic progression is a sequence of numbers in which each term after the first is found by adding a fixed number to the previous term. A geometric progression, on the other hand, is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed number.

Q: How can we use geometric progressions in real-life situations?

A: Geometric progressions have many real-life applications, such as:

  • Compound interest: When money is invested at a fixed interest rate, the interest earned is a geometric progression.
  • Population growth: The population of a country or city can be modeled using a geometric progression.
  • Sound waves: The amplitude of a sound wave can be modeled using a geometric progression.

Q: What are some common mistakes to avoid when working with geometric progressions?

A: Some common mistakes to avoid when working with geometric progressions include:

  • Assuming that the common ratio is always positive. In some cases, the common ratio can be negative.
  • Not checking for convergence or divergence of the sequence.
  • Not using the correct formula for the nth term.

Q: How can we use technology to help us work with geometric progressions?

A: There are many online tools and software programs that can help us work with geometric progressions, such as:

  • Graphing calculators: These can be used to visualize the sequence and calculate the nth term.
  • Spreadsheets: These can be used to create a table of values and calculate the nth term.
  • Online calculators: These can be used to calculate the nth term and other properties of the sequence.

Conclusion

In conclusion, geometric progressions are an important concept in mathematics that have many real-life applications. By understanding the formula for the nth term and the common ratio, we can use geometric progressions to model and analyze a wide range of phenomena.

Answer Key

  • Q1: A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Q2: The formula for the nth term of a geometric progression is given by an = ar^(n-1).
  • Q3: To determine the common ratio of a geometric progression, we can divide any term by its previous term.
  • Q4: An arithmetic progression is a sequence of numbers in which each term after the first is found by adding a fixed number to the previous term.
  • Q5: Geometric progressions have many real-life applications, such as compound interest, population growth, and sound waves.
  • Q6: Some common mistakes to avoid when working with geometric progressions include assuming that the common ratio is always positive, not checking for convergence or divergence of the sequence, and not using the correct formula for the nth term.
  • Q7: There are many online tools and software programs that can help us work with geometric progressions, such as graphing calculators, spreadsheets, and online calculators.

References

Related Topics