Four Students Wrote Sequences In Math Class.Angela: \[$-6, -9, -12, -15, \ldots\$\]Bradley: \[$-2, -6, -12, -24, \ldots\$\]Carter: \[$-1, -3, -9, -27, \ldots\$\]Dominique: \[$-1, -3, -9, -81, \ldots\$\]Which Student

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Introduction

In a math class, four students, Angela, Bradley, Carter, and Dominique, were asked to write sequences. The sequences they wrote are as follows:

  • Angela: {-6, -9, -12, -15, \ldots$}$
  • Bradley: {-2, -6, -12, -24, \ldots$}$
  • Carter: {-1, -3, -9, -27, \ldots$}$
  • Dominique: {-1, -3, -9, -81, \ldots$}$

The task is to 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. In other words, if the first term is aa, then the second term is arar, the third term is ar2ar^2, and so on, where rr is the common ratio.

Analyzing Angela's Sequence

Angela's sequence is {-6, -9, -12, -15, \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.

  • −9−6=32\frac{-9}{-6} = \frac{3}{2}
  • −12−9=43\frac{-12}{-9} = \frac{4}{3}
  • −15−12=54\frac{-15}{-12} = \frac{5}{4}

As we can see, the common ratio is not the same for each pair of consecutive terms. Therefore, Angela's sequence is not a geometric sequence.

Analyzing Bradley's Sequence

Bradley's sequence is {-2, -6, -12, -24, \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.

  • −6−2=3\frac{-6}{-2} = 3
  • −12−6=2\frac{-12}{-6} = 2
  • −24−12=2\frac{-24}{-12} = 2

As we can see, the common ratio is not the same for each pair of consecutive terms. However, we can see that the common ratio is 2 for the last two terms. Therefore, Bradley's sequence is not a geometric sequence.

Analyzing Carter's Sequence

Carter's sequence is {-1, -3, -9, -27, \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.

  • −3−1=3\frac{-3}{-1} = 3
  • −9−3=3\frac{-9}{-3} = 3
  • −27−9=3\frac{-27}{-9} = 3

As we can see, the common ratio is the same for each pair of consecutive terms. Therefore, Carter's sequence is a geometric sequence.

Analyzing Dominique's Sequence

Dominique's sequence is {-1, -3, -9, -81, \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.

  • −3−1=3\frac{-3}{-1} = 3
  • −9−3=3\frac{-9}{-3} = 3
  • −81−9=9\frac{-81}{-9} = 9

As we can see, the common ratio is not the same for each pair of consecutive terms. However, we can see that the common ratio is 3 for the first two terms and 9 for the last two terms. Therefore, Dominique's sequence is not a geometric sequence.

Conclusion

In conclusion, only Carter's sequence is a geometric sequence. The common ratio for Carter's sequence is 3.

What is the Common Ratio for Carter's Sequence?

The common ratio for Carter's sequence is 3.

Why is Carter's Sequence a Geometric Sequence?

Carter's sequence is a geometric sequence because each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio is 3.

What is the Formula for a Geometric Sequence?

The formula for a geometric sequence is an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, where ana_n is the nth term, a1a_1 is the first term, rr is the common ratio, and nn is the term number.

How Can We Use the Formula for a Geometric Sequence?

We can use the formula for a geometric sequence to find any term in the sequence. For example, if we want to find the 5th term in Carter's sequence, we can plug in the values a1=−1a_1 = -1, r=3r = 3, and n=5n = 5 into the formula.

What is the 5th Term in Carter's Sequence?

Using the formula for a geometric sequence, we get:

a5=−1⋅35−1=−1⋅34=−1⋅81=−81a_5 = -1 \cdot 3^{5-1} = -1 \cdot 3^4 = -1 \cdot 81 = -81

Therefore, the 5th term in Carter's sequence is -81.

What is the Sum of the First n Terms of a Geometric Sequence?

The sum of the first n terms of a geometric sequence is given by the formula:

Sn=a1(1−rn)1−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 number of terms.

How Can We Use the Formula for the Sum of a Geometric Sequence?

We can use the formula for the sum of a geometric sequence to find the sum of the first n terms of a geometric sequence. For example, if we want to find the sum of the first 5 terms in Carter's sequence, we can plug in the values a1=−1a_1 = -1, r=3r = 3, and n=5n = 5 into the formula.

What is the Sum of the First 5 Terms in Carter's Sequence?

Using the formula for the sum of a geometric sequence, we get:

S5=−1(1−35)1−3=−1(1−243)−2=−1(−242)−2=242−2=−121S_5 = \frac{-1(1-3^5)}{1-3} = \frac{-1(1-243)}{-2} = \frac{-1(-242)}{-2} = \frac{242}{-2} = -121

Therefore, the sum of the first 5 terms in Carter's sequence is -121.

What is the Product of the First n Terms of a Geometric Sequence?

The product of the first n terms of a geometric sequence is given by the formula:

Pn=a1⋅rn−1P_n = a_1 \cdot r^{n-1}

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

How Can We Use the Formula for the Product of a Geometric Sequence?

We can use the formula for the product of a geometric sequence to find the product of the first n terms of a geometric sequence. For example, if we want to find the product of the first 5 terms in Carter's sequence, we can plug in the values a1=−1a_1 = -1, r=3r = 3, and n=5n = 5 into the formula.

What is the Product of the First 5 Terms in Carter's Sequence?

Using the formula for the product of a geometric sequence, we get:

P5=−1⋅35−1=−1⋅34=−1⋅81=−81P_5 = -1 \cdot 3^{5-1} = -1 \cdot 3^4 = -1 \cdot 81 = -81

Therefore, the product of the first 5 terms in Carter's sequence is -81.

Conclusion

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

A: To determine if a sequence is a geometric sequence, you need to find the common ratio by dividing each term by the previous term. If the common ratio 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 an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}, where ana_n is the nth term, a1a_1 is the first term, rr is the common ratio, and nn is the term number.

Q: How can I use the formula for a geometric sequence?

A: You can use the formula for a geometric sequence to find any term in the sequence. Simply plug in the values for a1a_1, rr, and nn into the formula.

Q: What is the sum of the first n terms of a geometric sequence?

A: The sum of the first n terms of a geometric sequence is given by the formula:

Sn=a1(1−rn)1−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 number of terms.

Q: How can I use the formula for the sum of a geometric sequence?

A: You can use the formula for the sum of a geometric sequence to find the sum of the first n terms of a geometric sequence. Simply plug in the values for a1a_1, rr, and nn into the formula.

Q: What is the product of the first n terms of a geometric sequence?

A: The product of the first n terms of a geometric sequence is given by the formula:

Pn=a1⋅rn−1P_n = a_1 \cdot r^{n-1}

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

Q: How can I use the formula for the product of a geometric sequence?

A: You can use the formula for the product of a geometric sequence to find the product of the first n terms of a geometric sequence. Simply plug in the values for a1a_1, rr, and nn into the formula.

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

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

  • Compound interest: Geometric sequences can be used to calculate the future value of an investment with compound interest.
  • 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.
  • Art: Geometric sequences can be used to create intricate patterns and designs.

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

A: You can use geometric sequences in your everyday life by applying the concepts of geometric sequences to real-world problems. For example, you can use geometric sequences to calculate the future value of an investment, or to model the growth of a population.

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

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

  • Assuming that a sequence is a geometric sequence without checking the common ratio.
  • Using the wrong formula for a geometric sequence.
  • Not checking the domain of the formula for a geometric sequence.
  • Not considering the possibility of a zero or negative common ratio.

Q: How can I practice working with geometric sequences?

A: You can practice working with geometric sequences by:

  • Solving problems involving geometric sequences.
  • Creating your own geometric sequences and calculating the common ratio.
  • Using online resources and calculators to practice working with geometric sequences.
  • Joining a study group or working with a tutor to practice working with geometric sequences.