Given The Sequence: $\[ -2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots \\]Which Formula Can Be Used To Describe The Sequence?A. $f(x+1) = -2 F(x$\] B. $f(x+1) = -\frac{1}{2} F(x$\]

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Introduction

In mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. Given a sequence, identifying a formula that describes it can be a challenging but rewarding task. In this article, we will explore a sequence of numbers and attempt to find a formula that can be used to describe it.

The Given Sequence

The sequence provided is:

βˆ’223,βˆ’513,βˆ’1023,βˆ’2113,βˆ’4223,…{-2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots}

At first glance, the sequence appears to be a list of numbers with no apparent pattern. However, upon closer inspection, we can see that each term is obtained by multiplying the previous term by a certain factor.

Identifying the Pattern

Let's examine the differences between consecutive terms:

βˆ’513βˆ’(βˆ’223)=βˆ’213{-5 \frac{1}{3} - (-2 \frac{2}{3}) = -2 \frac{1}{3}}

βˆ’1023βˆ’(βˆ’513)=βˆ’513{-10 \frac{2}{3} - (-5 \frac{1}{3}) = -5 \frac{1}{3}}

βˆ’2113βˆ’(βˆ’1023)=βˆ’1023{-21 \frac{1}{3} - (-10 \frac{2}{3}) = -10 \frac{2}{3}}

βˆ’4223βˆ’(βˆ’2113)=βˆ’2113{-42 \frac{2}{3} - (-21 \frac{1}{3}) = -21 \frac{1}{3}}

We can see that the differences between consecutive terms are not constant, but they are all multiples of a certain number. Specifically, the differences are all multiples of βˆ’313-3 \frac{1}{3}.

Finding the Formula

Now that we have identified the pattern, we can attempt to find a formula that describes the sequence. Let's consider the following recurrence relation:

f(x+1)=βˆ’12f(x)f(x+1) = -\frac{1}{2} f(x)

This formula states that each term in the sequence is obtained by multiplying the previous term by βˆ’12-\frac{1}{2}. We can test this formula by applying it to the first few terms of the sequence:

f(1)=βˆ’223f(1) = -2 \frac{2}{3} f(2)=βˆ’12f(1)=βˆ’12(βˆ’223)=βˆ’513f(2) = -\frac{1}{2} f(1) = -\frac{1}{2} (-2 \frac{2}{3}) = -5 \frac{1}{3} f(3)=βˆ’12f(2)=βˆ’12(βˆ’513)=βˆ’1023f(3) = -\frac{1}{2} f(2) = -\frac{1}{2} (-5 \frac{1}{3}) = -10 \frac{2}{3}

As we can see, the formula f(x+1)=βˆ’12f(x)f(x+1) = -\frac{1}{2} f(x) produces the correct terms of the sequence.

Conclusion

In this article, we have explored a sequence of numbers and attempted to find a formula that describes it. By identifying the pattern in the sequence and using a recurrence relation, we were able to find a formula that produces the correct terms of the sequence. The formula f(x+1)=βˆ’12f(x)f(x+1) = -\frac{1}{2} f(x) is a simple and elegant solution that captures the underlying structure of the sequence.

Discussion

The sequence provided is a classic example of a geometric sequence, where each term is obtained by multiplying the previous term by a fixed factor. The formula f(x+1)=βˆ’12f(x)f(x+1) = -\frac{1}{2} f(x) is a common recurrence relation for geometric sequences, where the factor is βˆ’12-\frac{1}{2}.

References

  • [1] "Sequences and Series" by Michael Sullivan
  • [2] "Discrete Mathematics" by Kenneth H. Rosen

Additional Resources

  • [1] Khan Academy: Sequences and Series
  • [2] MIT OpenCourseWare: Discrete Mathematics and Its Applications

Final Thoughts

Q&A: Unraveling the Mysteries of the Sequence

In our previous article, we explored a sequence of numbers and attempted to find a formula that describes it. We discovered that the sequence is a geometric sequence with a common ratio of βˆ’12-\frac{1}{2}. In this article, we will answer some of the most frequently asked questions about the sequence and provide additional insights into its properties.

Q: What is a geometric sequence?

A geometric sequence is a type of sequence where each term is obtained by multiplying the previous term by a fixed factor, known as the common ratio.

Q: How do I determine the common ratio of a geometric sequence?

To determine the common ratio of a geometric sequence, you can divide each term by the previous term. In the case of the sequence we explored, the common ratio is βˆ’12-\frac{1}{2}.

Q: What is the formula for a geometric sequence?

The formula for a geometric sequence is:

an=a1β‹…rnβˆ’1a_n = a_1 \cdot r^{n-1}

where ana_n is the nth term of the sequence, a1a_1 is the first term, rr is the common ratio, and nn is the term number.

Q: How do I use the formula to find the nth term of a geometric sequence?

To use the formula to find the nth term of a geometric sequence, you need to know the first term and the common ratio. You can then plug these values into the formula and calculate the nth term.

Q: What are some real-world applications of geometric sequences?

Geometric sequences have many real-world applications, including:

  • Compound interest: Geometric sequences can be used to model the growth of an investment over time.
  • Population growth: Geometric sequences can be used to model the growth of a population over time.
  • Music: Geometric sequences can be used to create musical patterns and rhythms.

Q: Can I use the formula to find the sum of a geometric sequence?

Yes, you can use the formula to find the sum of a geometric sequence. The formula for the sum of a geometric sequence is:

Sn=a1(1βˆ’rn)1βˆ’rS_n = \frac{a_1(1-r^n)}{1-r}

where SnS_n is the sum of the first n terms, a1a_1 is the first term, rr is the common ratio, and nn is the term number.

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

Some common mistakes to avoid when working with geometric sequences include:

  • Assuming that the sequence is arithmetic instead of geometric.
  • Failing to identify the common ratio.
  • Using the wrong formula to find the nth term or the sum of the sequence.

Conclusion

In this article, we have answered some of the most frequently asked questions about geometric sequences and provided additional insights into their properties. We hope that this article has been helpful in unraveling the mysteries of the sequence and has provided you with a deeper understanding of geometric sequences.

Additional Resources

  • [1] Khan Academy: Sequences and Series
  • [2] MIT OpenCourseWare: Discrete Mathematics and Its Applications
  • [3] Wolfram MathWorld: Geometric Sequence

Final Thoughts

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena. By understanding the properties of geometric sequences, you can unlock the secrets of the sequence and gain a deeper appreciation for the beauty of mathematics.