{ \begin{array}{|c|c|} \hline x & \left(\frac{2}{3}\right)^x \\ \hline -1 & \frac{3}{2} \\ \hline 0 & D \\ \hline 2 & E \\ \hline 4 & F \\ \hline \end{array} \}$Calculate The Values:${ d = \square \}$\[ e =

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Introduction

In this article, we will explore a geometric sequence and calculate the values of its terms. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The common ratio in this sequence is 23\frac{2}{3}.

Understanding the Sequence

The given sequence is:

xx (23)x\left(\frac{2}{3}\right)^x
-1 32\frac{3}{2}
0 dd
2 ee
4 ff

We are asked to calculate the values of dd, ee, and ff.

Calculating dd

To calculate the value of dd, we need to find the term when x=0x = 0. In a geometric sequence, the term when x=0x = 0 is always 1, because any number raised to the power of 0 is 1. Therefore, d=1d = 1.

Calculating ee

To calculate the value of ee, we need to find the term when x=2x = 2. We can do this by multiplying the previous term by the common ratio. The previous term is 32\frac{3}{2}, and the common ratio is 23\frac{2}{3}. Therefore, e=32×23=1e = \frac{3}{2} \times \frac{2}{3} = 1.

Calculating ff

To calculate the value of ff, we need to find the term when x=4x = 4. We can do this by multiplying the previous term by the common ratio. The previous term is 11, and the common ratio is 23\frac{2}{3}. Therefore, f=1×23=23f = 1 \times \frac{2}{3} = \frac{2}{3}.

Conclusion

In this article, we calculated the values of dd, ee, and ff in a geometric sequence. We found that d=1d = 1, e=1e = 1, and f=23f = \frac{2}{3}. This demonstrates the power of geometric sequences and how they can be used to model real-world phenomena.

Geometric Sequences in Real-World Applications

Geometric sequences have many real-world applications, including:

  • Finance: Geometric sequences can be used to model the growth of investments over time.
  • Biology: Geometric sequences can be used to model the growth of populations over time.
  • Computer Science: Geometric sequences can be used to model the growth of data in computer systems.

Examples of Geometric Sequences

Here are some examples of geometric sequences:

  • Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, ...
  • Geometric Sequence with Common Ratio 2: 1, 2, 4, 8, 16, ...
  • Geometric Sequence with Common Ratio 3: 1, 3, 9, 27, 81, ...

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena. They have many applications in finance, biology, and computer science, and can be used to model the growth of investments, populations, and data over time. By understanding geometric sequences, we can gain a deeper understanding of the world around us.

References

  • "Geometric Sequences" by Math Open Reference
  • "Geometric Sequences and Series" by Khan Academy
  • "Geometric Sequences in Finance" by Investopedia

Further Reading

  • "Geometric Sequences and Series" by Michael Corral
  • "Geometric Sequences in Biology" by Science Daily
  • "Geometric Sequences in Computer Science" by GeeksforGeeks
    Geometric Sequences Q&A =========================

Introduction

In this article, we will answer some frequently asked questions about geometric sequences. Geometric sequences are a type of sequence where 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 a geometric sequence?

A: A geometric sequence is a sequence of numbers where 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 common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the number that is multiplied by each term to get the next term. For example, in the sequence 2, 6, 18, 54, ..., the common ratio is 3.

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

A: To find the nth term of a geometric sequence, you can use the formula:

an = a1 × r^(n-1)

where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is:

S = a1 × (1 - r^n) / (1 - r)

where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

Q: How do I determine if a sequence is geometric?

A: To determine if a sequence is geometric, you can check if each term after the first is found by multiplying the previous term by a fixed, non-zero number. You can also use the formula for the nth term to see if it matches the sequence.

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

A: Geometric sequences have many real-world applications, including:

  • Finance: Geometric sequences can be used to model the growth of investments over time.
  • Biology: Geometric sequences can be used to model the growth of populations over time.
  • Computer Science: Geometric sequences can be used to model the growth of data in computer systems.

Q: How do I find the sum of an infinite geometric sequence?

A: To find the sum of an infinite geometric sequence, you can use the formula:

S = a1 / (1 - r)

where S is the sum, a1 is the first term, and r is the common ratio.

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

A: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number. An arithmetic sequence is a sequence where each term after the first is found by adding a fixed number to the previous term.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena. They have many applications in finance, biology, and computer science, and can be used to model the growth of investments, populations, and data over time. By understanding geometric sequences, we can gain a deeper understanding of the world around us.

References

  • "Geometric Sequences" by Math Open Reference
  • "Geometric Sequences and Series" by Khan Academy
  • "Geometric Sequences in Finance" by Investopedia

Further Reading

  • "Geometric Sequences and Series" by Michael Corral
  • "Geometric Sequences in Biology" by Science Daily
  • "Geometric Sequences in Computer Science" by GeeksforGeeks