A Geometric Sequence Begins With ${ 72, 36, 18, 9, \ldots\$} .Which Option Below Represents The Formula For The Sequence?A. { F(n) = 72(2)^{n-1}$}$B. { F(n) = 72(2)^{n+1}$}$C. { F(n) = 72(0.5)^{n-1}$}$D.
Introduction
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 explore the concept of geometric sequences and how to represent them using a formula. We will also examine a specific geometric sequence and determine the correct formula for the 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 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.
Example of a Geometric Sequence
The given geometric sequence is:
72, 36, 18, 9, ...
To find the common ratio, we can divide each term by the previous term:
36 ÷ 72 = 0.5 18 ÷ 36 = 0.5 9 ÷ 18 = 0.5
The common ratio is 0.5. Now, we can use the formula for a geometric sequence to find the nth term:
a_n = 72 * (0.5)^(n-1)
Representing the Sequence Using a Formula
Now that we have found the common ratio, we can represent the sequence using a formula. The formula for the sequence is:
f(n) = 72(0.5)^(n-1)
This formula represents the nth term of the sequence.
Analyzing the Options
Let's analyze the options given:
A. f(n) = 72(2)^(n-1) B. f(n) = 72(2)^(n+1) C. f(n) = 72(0.5)^(n-1) D. ...
Option A is incorrect because the common ratio is 0.5, not 2. Option B is also incorrect because the common ratio is 0.5, not 2. Option C is the correct formula for the sequence.
Conclusion
In conclusion, a geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. We have examined a specific geometric sequence and determined the correct formula for the sequence. The formula for the sequence is f(n) = 72(0.5)^(n-1).
Common Ratio
The common ratio is a crucial component of a geometric sequence. It determines how each term is related to the previous term. In the given sequence, the common ratio is 0.5.
Geometric Sequence Formula
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.
Example of a Geometric Sequence Formula
The formula for the given sequence is:
f(n) = 72(0.5)^(n-1)
This formula represents the nth term of the sequence.
Geometric Sequence Applications
Geometric sequences have numerous applications in mathematics, science, and engineering. They are used to model population growth, financial investments, and electrical circuits, among other things.
Geometric Sequence Formula Derivation
The formula for a geometric sequence can be derived by examining the relationship between consecutive terms. By dividing each term by the previous term, we can find the common ratio.
Geometric Sequence Formula Properties
The formula for a geometric sequence has several properties, including:
- The formula is recursive, meaning that each term is defined in terms of the previous term.
- The formula is exponential, meaning that each term is a power of the common ratio.
- The formula is additive, meaning that the sum of the terms is a geometric series.
Geometric Sequence Formula Examples
The formula for a geometric sequence can be used to find the nth term of a sequence. For example, if we want to find the 5th term of the sequence 2, 4, 8, 16, ..., we can use the formula:
f(n) = 2(2)^(n-1)
Plugging in n = 5, we get:
f(5) = 2(2)^(5-1) = 2(2)^4 = 2(16) = 32
Therefore, the 5th term of the sequence is 32.
Geometric Sequence Formula Applications
The formula for a geometric sequence has numerous applications in mathematics, science, and engineering. For example, it can be used to model population growth, financial investments, and electrical circuits.
Geometric Sequence Formula Derivation
The formula for a geometric sequence can be derived by examining the relationship between consecutive terms. By dividing each term by the previous term, we can find the common ratio.
Geometric Sequence Formula Properties
The formula for a geometric sequence has several properties, including:
- The formula is recursive, meaning that each term is defined in terms of the previous term.
- The formula is exponential, meaning that each term is a power of the common ratio.
- The formula is additive, meaning that the sum of the terms is a geometric series.
Geometric Sequence Formula Examples
The formula for a geometric sequence can be used to find the nth term of a sequence. For example, if we want to find the 5th term of the sequence 2, 4, 8, 16, ..., we can use the formula:
f(n) = 2(2)^(n-1)
Plugging in n = 5, we get:
f(5) = 2(2)^(5-1) = 2(2)^4 = 2(16) = 32
Therefore, the 5th term of the sequence is 32.
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 number called the common ratio.
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 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. For example, if the sequence is 2, 4, 8, 16, ..., you can divide each term by the previous term to find the common ratio:
4 ÷ 2 = 2 8 ÷ 4 = 2 16 ÷ 8 = 2
The common ratio is 2.
Q: How do I use the formula for a geometric sequence to find the nth term?
A: To use the formula for a geometric sequence to find the nth term, you need to plug in the values of a_1, r, and n into the formula. For example, if the sequence is 2, 4, 8, 16, ..., and you want to find the 5th term, you can use the formula:
f(n) = 2(2)^(n-1)
Plugging in n = 5, you get:
f(5) = 2(2)^(5-1) = 2(2)^4 = 2(16) = 32
Therefore, the 5th term of the sequence is 32.
Q: What are some real-world applications of geometric sequences?
A: Geometric sequences have numerous real-world applications, including:
- Modeling population growth
- Financial investments
- Electrical circuits
- Music and art
Q: Can I use the formula for a geometric sequence to find the sum of a geometric series?
A: Yes, you can use the formula for a geometric sequence to find the sum of a geometric series. The formula for the sum of a geometric series is:
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: How do I determine if a sequence is geometric?
A: To determine if a sequence is geometric, you can check if each term is found by multiplying the previous term by a fixed number called the common ratio. If this is the case, then the sequence is geometric.
Q: Can I use the formula for a geometric sequence to find the nth term of a sequence with a negative common ratio?
A: Yes, you can use the formula for a geometric sequence to find the nth term of a sequence with a negative common ratio. However, you need to be careful when working with negative numbers, as the formula may produce complex numbers.
Q: How do I find the nth term of a sequence with a common ratio that is not an integer?
A: To find the nth term of a sequence with a common ratio that is not an integer, you can use the formula for a geometric sequence and plug in the values of a_1, r, and n. However, you may need to use a calculator or computer to evaluate the expression.
Q: Can I use the formula for a geometric sequence to find the sum of a geometric series with a negative common ratio?
A: Yes, you can use the formula for a geometric sequence to find the sum of a geometric series with a negative common ratio. However, you need to be careful when working with negative numbers, as the formula may produce complex numbers.
Conclusion
In conclusion, geometric sequences are a type of sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. We have examined the formula for a geometric sequence and how to use it to find the nth term and the sum of a geometric series. We have also discussed some real-world applications of geometric sequences and how to determine if a sequence is geometric.