Which Sequences Are Geometric? Check All That Apply.A. \[$-2, -4, -6, -8, -10, \ldots\$\]B. \[$16, -8, 4, -2, 1\$\]C. \[$-15, -18, -21.6, -25.92, -31.104, \ldots\$\]D. \[$4, 10.5, 17, 23.5, 30, \ldots\$\]E. \[$625,
Which Sequences are Geometric? Check All That Apply
In mathematics, 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. Geometric sequences are commonly used in finance, physics, and engineering to model growth or decay over time. In this article, we will examine five given sequences and determine which ones are geometric.
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.
Sequence A: [-2, -4, -6, -8, -10, ...]
To determine if Sequence A is geometric, we need to find the common ratio between consecutive terms.
-2 * (-2) = 4 -4 * (-2) = 8 -6 * (-2) = 12 -8 * (-2) = 16 -10 * (-2) = 20
As we can see, the common ratio between consecutive terms is not constant, and the sequence does not follow the formula for a geometric sequence. Therefore, Sequence A is not geometric.
Sequence B: [16, -8, 4, -2, 1]
To determine if Sequence B is geometric, we need to find the common ratio between consecutive terms.
16 * (-1/2) = -8 -8 * (-1/2) = 4 4 * (-1/2) = -2 -2 * (-1/2) = 1
As we can see, the common ratio between consecutive terms is constant, and the sequence follows the formula for a geometric sequence. Therefore, Sequence B is geometric.
Sequence C: [-15, -18, -21.6, -25.92, -31.104, ...]
To determine if Sequence C is geometric, we need to find the common ratio between consecutive terms.
-15 * (-1.2) = -18 -18 * (-1.2) = -21.6 -21.6 * (-1.2) = -25.92 -25.92 * (-1.2) = -31.104
As we can see, the common ratio between consecutive terms is constant, and the sequence follows the formula for a geometric sequence. Therefore, Sequence C is geometric.
Sequence D: [4, 10.5, 17, 23.5, 30, ...]
To determine if Sequence D is geometric, we need to find the common ratio between consecutive terms.
4 * (2.625) = 10.5 10.5 * (2.625) = 17 17 * (2.625) = 23.5 23.5 * (2.625) = 30
As we can see, the common ratio between consecutive terms is not constant, and the sequence does not follow the formula for a geometric sequence. Therefore, Sequence D is not geometric.
Sequence E: [625, 125, 25, 5, 1]
To determine if Sequence E is geometric, we need to find the common ratio between consecutive terms.
625 * (1/5) = 125 125 * (1/5) = 25 25 * (1/5) = 5 5 * (1/5) = 1
As we can see, the common ratio between consecutive terms is constant, and the sequence follows the formula for a geometric sequence. Therefore, Sequence E is geometric.
Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial in various fields such as finance, physics, and engineering. In our previous article, we examined five given sequences and determined which ones are geometric. 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 can divide any term by its previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can find the common ratio by dividing 6 by 2, which is 3.
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: How do I determine if a sequence is geometric?
A: To determine if a sequence is geometric, you can check if the ratio between consecutive terms is constant. If the ratio is constant, then the sequence is geometric.
Q: What are some real-world applications of geometric sequences?
A: Geometric sequences have many real-world applications, including:
- Finance: Geometric sequences are used to calculate compound interest and investment returns.
- Physics: Geometric sequences are used to model the motion of objects under constant acceleration.
- Engineering: Geometric sequences are used to design and analyze electrical circuits and mechanical systems.
Q: Can a geometric sequence have a negative common ratio?
A: Yes, a geometric sequence can have a negative common ratio. For example, the sequence 2, -6, 18, -54, ... has a common ratio of -3.
Q: Can a geometric sequence have a common ratio of 1?
A: Yes, a geometric sequence can have a common ratio of 1. For example, the sequence 2, 2, 2, 2, ... has a common ratio of 1.
Q: What is the sum of an infinite geometric sequence?
A: The sum of an infinite geometric sequence is given by the formula:
S = a_1 / (1 - r)
where S is the sum, a_1 is the first term, and r is the common ratio.
In conclusion, geometric sequences are a fundamental concept in mathematics, and understanding them is crucial in various fields. We hope that this article has answered some of the frequently asked questions about geometric sequences and has provided a better understanding of this important mathematical concept.