Four Students Wrote Sequences During Math Class. Identify Which Student Wrote A Geometric Sequence.Andre:1. { -\frac 3}{4}, \frac{3}{8}, -\frac{3}{16}, -\frac{3}{32}, \ldots$}$Brenda 1. [$\frac{3 {4}, -\frac{3}{8}, \frac{3}{16},

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Introduction

In mathematics, a geometric sequence is 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. In this article, we will analyze the sequences written by four students during math class and identify which student wrote a geometric sequence.

What is a Geometric Sequence?

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 general formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Analyzing the Sequences

Let's analyze the sequences written by the four students:

Andre's Sequence

Andre's sequence is:

  1. {-\frac{3}{4}, \frac{3}{8}, -\frac{3}{16}, -\frac{3}{32}, \ldots$}$

To determine if this is a geometric sequence, we need to find the common ratio. We can do this by dividing each term by the previous term:

โˆ’34รท38=โˆ’34ร—83=โˆ’2{-\frac{3}{4} \div \frac{3}{8} = -\frac{3}{4} \times \frac{8}{3} = -2}

38รทโˆ’316=38ร—โˆ’163=โˆ’2{\frac{3}{8} \div -\frac{3}{16} = \frac{3}{8} \times -\frac{16}{3} = -2}

โˆ’316รทโˆ’332=โˆ’316ร—โˆ’323=2{-\frac{3}{16} \div -\frac{3}{32} = -\frac{3}{16} \times -\frac{32}{3} = 2}

The common ratio is not constant, so Andre's sequence is not a geometric sequence.

Brenda's Sequence

Brenda's sequence is:

  1. {\frac{3}{4}, -\frac{3}{8}, \frac{3}{16}, \frac{3}{32}, \ldots$}$

To determine if this is a geometric sequence, we need to find the common ratio. We can do this by dividing each term by the previous term:

34รทโˆ’38=34ร—โˆ’83=โˆ’2{\frac{3}{4} \div -\frac{3}{8} = \frac{3}{4} \times -\frac{8}{3} = -2}

โˆ’38รท316=โˆ’38ร—163=โˆ’2{-\frac{3}{8} \div \frac{3}{16} = -\frac{3}{8} \times \frac{16}{3} = -2}

316รท332=316ร—323=2{\frac{3}{16} \div \frac{3}{32} = \frac{3}{16} \times \frac{32}{3} = 2}

The common ratio is not constant, so Brenda's sequence is not a geometric sequence.

Charlie's Sequence

Charlie's sequence is:

  1. ${2, 6, 18, 54, \ldots\$}

To determine if this is a geometric sequence, we need to find the common ratio. We can do this by dividing each term by the previous term:

6รท2=3{6 \div 2 = 3}

18รท6=3{18 \div 6 = 3}

54รท18=3{54 \div 18 = 3}

The common ratio is constant, so Charlie's sequence is a geometric sequence.

David's Sequence

David's sequence is:

  1. {-2, 6, -18, 54, \ldots$}$

To determine if this is a geometric sequence, we need to find the common ratio. We can do this by dividing each term by the previous term:

6รทโˆ’2=โˆ’3{6 \div -2 = -3}

โˆ’18รท6=โˆ’3{-18 \div 6 = -3}

54รทโˆ’18=โˆ’3{54 \div -18 = -3}

The common ratio is constant, so David's sequence is a geometric sequence.

Conclusion

In conclusion, only two students, Charlie and David, wrote geometric sequences during math class. Charlie's sequence has a common ratio of 3, while David's sequence has a common ratio of -3. The other two students, Andre and Brenda, did not write geometric sequences.

What is the Importance of Geometric Sequences?

Geometric sequences are important in mathematics because they can be used to model real-world situations, such as population growth, financial investments, and electrical circuits. They are also used in many mathematical formulas and theorems, such as the formula for the sum of a geometric series.

Real-World Applications of Geometric Sequences

Geometric sequences have many real-world applications, including:

  • Population growth: Geometric sequences can be used to model population growth, where the population grows at a constant rate.
  • Financial investments: Geometric sequences can be used to calculate the future value of an investment, where the interest rate is constant.
  • Electrical circuits: Geometric sequences can be used to model the behavior of electrical circuits, where the voltage and current are related by a geometric sequence.

Conclusion

Q: What is a geometric sequence?

A: A geometric sequence is 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: How do I find the common ratio of a geometric sequence?

A: To find the common ratio of a geometric sequence, you can divide each term by the previous term. If the result is the same for each pair of consecutive terms, then the sequence is a geometric sequence.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

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 called the common ratio. An arithmetic sequence, on the other hand, is a sequence where each term after the first is found by adding a fixed number called the common difference.

Q: Can a geometric sequence have a common ratio of 1?

A: Yes, a geometric sequence can have a common ratio of 1. In this case, the sequence is a constant sequence, where each term is the same as the previous term.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. If the common ratio is 0, then the sequence is a constant sequence, where each term is 0.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence can be found using the formula:

S_n = a_1 * (1 - r^n) / (1 - r)

where S_n is the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.

Q: What is the product of a geometric sequence?

A: The product of a geometric sequence can be found using the formula:

P_n = a_1 * r^(n-1)

where P_n is the product of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.

Q: Can a geometric sequence be used to model real-world situations?

A: Yes, a geometric sequence can be used to model real-world situations, such as population growth, financial investments, and electrical circuits.

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

A: Some real-world applications of geometric sequences include:

  • Population growth: Geometric sequences can be used to model population growth, where the population grows at a constant rate.
  • Financial investments: Geometric sequences can be used to calculate the future value of an investment, where the interest rate is constant.
  • Electrical circuits: Geometric sequences can be used to model the behavior of electrical circuits, where the voltage and current are related by a geometric sequence.

Q: How can I use geometric sequences in my daily life?

A: You can use geometric sequences in your daily life by applying the concepts of geometric sequences to real-world situations, such as:

  • Investing: Geometric sequences can be used to calculate the future value of an investment, where the interest rate is constant.
  • Population growth: Geometric sequences can be used to model population growth, where the population grows at a constant rate.
  • Electrical circuits: Geometric sequences can be used to model the behavior of electrical circuits, where the voltage and current are related by a geometric sequence.

Conclusion

In conclusion, geometric sequences are an important concept in mathematics, with many real-world applications. By understanding geometric sequences, you can better model and analyze real-world situations, and make more informed decisions.