What Is The Common Ratio Of The Geometric Sequence Below?$625, 125, 25, 5, 1, \ldots$A. $\frac{1}{625}$B. $\frac{25}{125}$C. $\frac{625}{125}$D. 125
What is a Geometric Sequence?
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 means that if we know the first term and the common ratio, we can find any term in the sequence.
The Given Geometric Sequence
The given geometric sequence is: 625, 125, 25, 5, 1, ...
Finding the Common Ratio
To find the common ratio, we need to divide each term by the previous term. Let's start with the first two terms:
- 125 ÷ 625 = 1/5
- 25 ÷ 125 = 1/5
- 5 ÷ 25 = 1/5
- 1 ÷ 5 = 1/5
As we can see, the common ratio is 1/5 in each case. This means that to get from one term to the next, we multiply by 1/5.
Why is the Common Ratio Important?
The common ratio is an essential concept in geometric sequences because it helps us understand how the sequence grows or shrinks. If the common ratio is greater than 1, the sequence grows exponentially. If the common ratio is less than 1, the sequence shrinks exponentially.
Common Ratio in the Given Sequence
In the given sequence, the common ratio is 1/5. This means that each term is 1/5 of the previous term. For example, to get from 625 to 125, we multiply 625 by 1/5, which gives us 125.
Conclusion
In conclusion, the common ratio of the given geometric sequence is 1/5. This means that each term is 1/5 of the previous term. Understanding the common ratio is crucial in geometric sequences because it helps us understand how the sequence grows or shrinks.
Answer
The correct answer is A. 1/625 is not the correct answer, but 1/5 is the correct answer, however, the answer is not listed, but the closest answer is B. 25/125 which is 1/5.
Why is the Answer Not Listed?
The answer is not listed because the question is asking for the common ratio, which is 1/5, but the answer choices are not correct. The closest answer is B. 25/125, which is 1/5.
Final Thoughts
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, you need to divide each term by the previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can find the common ratio by dividing each term by the previous term: 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3. The common ratio is 3.
Q: What is the formula for finding the nth term of a geometric sequence?
A: The formula for finding the nth term of a geometric sequence is: 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: How do I determine if a sequence is geometric?
A: To determine if a sequence is geometric, you need to check if each term is obtained by multiplying the previous term by a fixed number. If this is the case, then the sequence is geometric.
Q: What is the difference between an arithmetic sequence and a geometric sequence?
A: An arithmetic sequence is a sequence where each term is obtained by adding a fixed number to the previous term. A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed number.
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 would be a constant sequence, where each term is the same.
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 would be a constant sequence, where each term is the same.
Q: How do I find the sum of a geometric sequence?
A: To find the sum of a geometric sequence, you can use 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: What is the application of geometric sequences in real-life?
A: Geometric sequences have many applications in real-life, such as:
- Compound interest: Geometric sequences can be used to calculate compound interest on an investment.
- Population growth: Geometric sequences can be used to model population growth.
- Music: Geometric sequences can be used to create musical patterns.
- Art: Geometric sequences can be used to create artistic patterns.
Q: Can I use a calculator to find the common ratio of a geometric sequence?
A: Yes, you can use a calculator to find the common ratio of a geometric sequence. Simply enter the terms of the sequence into the calculator and use the "r" function to find the common ratio.
Q: Can I use a spreadsheet to find the sum of a geometric sequence?
A: Yes, you can use a spreadsheet to find the sum of a geometric sequence. Simply enter the terms of the sequence into the spreadsheet and use 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.