Which Sequences Are Geometric Sequences? Check All That Apply.A. $4, 2, 1, \frac{1}{2}, \frac{1}{4}, \cdots$B. $-2, 3, -4, 5, -6$C. $2, 6, 18, 54, 162$D. $-4, -16, -64, -256, \cdots$E. $-2, -4, -12, -48, -240,

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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. This type of sequence is commonly used in various mathematical and real-world applications, such as finance, physics, and engineering. In this article, we will explore which sequences are geometric sequences and check all that apply.

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, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio.

Checking the Sequences

Let's check each of the given sequences to see if they are geometric sequences.

A. 4,2,1,12,14,⋯4, 2, 1, \frac{1}{2}, \frac{1}{4}, \cdots

To check if this sequence is a geometric sequence, we need to find the common ratio between consecutive terms. Let's calculate the ratio between the first two terms:

24=12\frac{2}{4} = \frac{1}{2}

Now, let's calculate the ratio between the second and third terms:

12=12\frac{1}{2} = \frac{1}{2}

Since the ratio between consecutive terms is the same, we can conclude that this sequence is a geometric sequence with a common ratio of 12\frac{1}{2}.

B. −2,3,−4,5,−6-2, 3, -4, 5, -6

To check if this sequence is a geometric sequence, we need to find the common ratio between consecutive terms. Let's calculate the ratio between the first two terms:

3−2=−32\frac{3}{-2} = -\frac{3}{2}

Now, let's calculate the ratio between the second and third terms:

−43=−43\frac{-4}{3} = -\frac{4}{3}

Since the ratio between consecutive terms is not the same, we can conclude that this sequence is not a geometric sequence.

C. 2,6,18,54,1622, 6, 18, 54, 162

To check if this sequence is a geometric sequence, we need to find the common ratio between consecutive terms. Let's calculate the ratio between the first two terms:

62=3\frac{6}{2} = 3

Now, let's calculate the ratio between the second and third terms:

186=3\frac{18}{6} = 3

Since the ratio between consecutive terms is the same, we can conclude that this sequence is a geometric sequence with a common ratio of 3.

D. −4,−16,−64,−256,⋯-4, -16, -64, -256, \cdots

To check if this sequence is a geometric sequence, we need to find the common ratio between consecutive terms. Let's calculate the ratio between the first two terms:

−16−4=4\frac{-16}{-4} = 4

Now, let's calculate the ratio between the second and third terms:

−64−16=4\frac{-64}{-16} = 4

Since the ratio between consecutive terms is the same, we can conclude that this sequence is a geometric sequence with a common ratio of 4.

E. −2,−4,−12,−48,−240-2, -4, -12, -48, -240

To check if this sequence is a geometric sequence, we need to find the common ratio between consecutive terms. Let's calculate the ratio between the first two terms:

−4−2=2\frac{-4}{-2} = 2

Now, let's calculate the ratio between the second and third terms:

−12−4=3\frac{-12}{-4} = 3

Since the ratio between consecutive terms is not the same, we can conclude that this sequence is not a geometric sequence.

Conclusion

In conclusion, the geometric sequences among the given options are:

  • A. 4,2,1,12,14,⋯4, 2, 1, \frac{1}{2}, \frac{1}{4}, \cdots
  • C. 2,6,18,54,1622, 6, 18, 54, 162
  • D. −4,−16,−64,−256,⋯-4, -16, -64, -256, \cdots

These sequences have a common ratio between consecutive terms, which is a characteristic of geometric sequences. The other sequences do not have a common ratio between consecutive terms, and therefore, are not geometric sequences.

Real-World Applications of Geometric Sequences

Geometric sequences have numerous real-world applications, such as:

  • Finance: Geometric sequences are used to calculate compound interest, where the interest is added to the principal amount at regular intervals.
  • Physics: Geometric sequences are used to describe the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
  • Engineering: Geometric sequences are used to design and analyze systems, such as electrical circuits or mechanical systems.

Tips for Identifying Geometric Sequences

To identify geometric sequences, follow these tips:

  • Look for a common ratio: Check if the ratio between consecutive terms is the same.
  • Check the ratio between terms: Calculate the ratio between consecutive terms to see if it is the same.
  • Use the formula: Use the formula for a geometric sequence to check if the sequence is geometric.

In the previous article, we explored what geometric sequences are and how to identify them. In this article, we will answer some frequently asked questions about geometric sequences.

Q: What is the difference between a geometric sequence and an arithmetic 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. An arithmetic sequence, on the other hand, is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

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

A: To find the common ratio of a geometric sequence, you need to divide each term by the previous term. For example, if the sequence is 2, 6, 18, 54, 162, you can find the common ratio by dividing each term by the previous term:

  • 6 ÷ 2 = 3
  • 18 ÷ 6 = 3
  • 54 ÷ 18 = 3
  • 162 ÷ 54 = 3

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 sum of a geometric sequence?

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

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 or not?

A: To determine if a sequence is geometric or not, you need to check if the ratio between consecutive terms is the same. If the ratio is the same, then the sequence is geometric. If the ratio is not the same, then the sequence is not geometric.

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

A: Geometric sequences have numerous real-world applications, such as:

  • Finance: Geometric sequences are used to calculate compound interest, where the interest is added to the principal amount at regular intervals.
  • Physics: Geometric sequences are used to describe the motion of objects, such as the trajectory of a projectile or the vibration of a spring.
  • Engineering: Geometric sequences are used to design and analyze systems, such as electrical circuits or mechanical systems.

Q: How do I use geometric sequences in finance?

A: Geometric sequences are used in finance to calculate compound interest, where the interest is added to the principal amount at regular intervals. For example, if you invest $100 at a 5% annual interest rate, the amount after one year will be:

$100 + ($100 * 0.05) = $105

After two years, the amount will be:

$105 + ($105 * 0.05) = $110.25

And so on.

Q: How do I use geometric sequences in physics?

A: Geometric sequences are used in physics to describe the motion of objects, such as the trajectory of a projectile or the vibration of a spring. For example, if a ball is thrown upwards with an initial velocity of 10 m/s, the height of the ball at any time t can be described by a geometric sequence:

h(t) = 10 * (1 - 0.5^t)

where h(t) is the height at time t.

Q: How do I use geometric sequences in engineering?

A: Geometric sequences are used in engineering to design and analyze systems, such as electrical circuits or mechanical systems. For example, if you want to design a circuit with a resistance of 10 ohms and a capacitance of 1 farad, the impedance of the circuit can be described by a geometric sequence:

Z = 10 * (1 - 0.5^t)

where Z is the impedance at time t.

By understanding geometric sequences and their applications, you can solve a wide range of problems in finance, physics, and engineering.