The First Four Terms Of A Geometric Sequence Are ${ 108, 36, 12, 4, \ldots\$} .What Is The Common Ratio?A. { -72$}$ B. { -3$}$ C. ${ 3\$} D. { \frac{1}{3}$}$
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 how to find the common ratio of a geometric sequence given its first four terms.
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_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.
The First Four Terms of the Geometric Sequence
The first four terms of the geometric sequence are given as:
108, 36, 12, 4, ...
Finding the Common Ratio
To find the common ratio, we can use the formula:
r = (a_n / a_(n-1))
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Let's use the first four terms to find the common ratio:
r = (36 / 108) = 1/3
r = (12 / 36) = 1/3
r = (4 / 12) = 1/3
As we can see, the common ratio is the same in all three cases, which is 1/3.
Conclusion
In this article, we have explored how to find the common ratio of a geometric sequence given its first four terms. We have used the formula r = (a_n / a_(n-1)) to find the common ratio, and we have seen that the common ratio is the same in all three cases, which is 1/3.
The Final Answer
The final answer is:
- D. {\frac{1}{3}$}$
Additional Information
- The common ratio can be positive or negative.
- The common ratio can be a fraction or a decimal.
- The common ratio can be found using the formula r = (a_n / a_(n-1)).
Example Problems
- Find the common ratio of the geometric sequence: 2, 6, 18, 54, ...
- Find the common ratio of the geometric sequence: 3, 9, 27, 81, ...
- Find the common ratio of the geometric sequence: 4, 16, 64, 256, ...
Solutions to Example Problems
- The common ratio of the geometric sequence 2, 6, 18, 54, ... is 3.
- The common ratio of the geometric sequence 3, 9, 27, 81, ... is 3.
- The common ratio of the geometric sequence 4, 16, 64, 256, ... is 4.
Tips and Tricks
- To find the common ratio, use the formula r = (a_n / a_(n-1)).
- Make sure to use the correct terms in the formula.
- Check your answer by plugging it back into the formula.
Common Ratio Formula
r = (a_n / a_(n-1))
Common Ratio Examples
- Find the common ratio of the geometric sequence: 2, 6, 18, 54, ...
- Find the common ratio of the geometric sequence: 3, 9, 27, 81, ...
- Find the common ratio of the geometric sequence: 4, 16, 64, 256, ...
Common Ratio Solutions
- The common ratio of the geometric sequence 2, 6, 18, 54, ... is 3.
- The common ratio of the geometric sequence 3, 9, 27, 81, ... is 3.
- The common ratio of the geometric sequence 4, 16, 64, 256, ... is 4.
Common Ratio 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, non-zero number called the common ratio.
Q: What is the formula for a geometric sequence?
A: 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.
Q: How do I find the common ratio of a geometric sequence?
A: To find the common ratio, you can use the formula:
r = (a_n / a_(n-1))
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Q: What if I don't know the first term of the sequence?
A: If you don't know the first term of the sequence, you can use the formula:
r = (a_n / a_(n-2))
where r is the common ratio, a_n is the nth term, and a_(n-2) is the (n-2)th term.
Q: Can the common ratio be negative?
A: Yes, the common ratio can be negative.
Q: Can the common ratio be a fraction or a decimal?
A: Yes, the common ratio can be a fraction or a decimal.
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:
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: What if I want to find the sum of the first n terms of a geometric sequence?
A: To find the sum of the first n terms of a geometric sequence, you can use the formula:
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 term number.
Q: Can I use a calculator to find the common ratio and the nth term of a geometric sequence?
A: Yes, you can use a calculator to find the common ratio and the nth term of a geometric sequence.
Q: What if I want to find the common ratio of a geometric sequence with a negative first term?
A: To find the common ratio of a geometric sequence with a negative first term, you can use the formula:
r = (-a_n / a_(n-1))
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Q: Can I use the formula for the sum of a geometric sequence to find the common ratio?
A: No, you cannot use the formula for the sum of a geometric sequence to find the common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a zero first term?
A: To find the common ratio of a geometric sequence with a zero first term, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the nth term of a geometric sequence to find the common ratio?
A: No, you cannot use the formula for the nth term of a geometric sequence to find the common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a negative common ratio?
A: To find the common ratio of a geometric sequence with a negative common ratio, you can use the formula:
r = (-a_n / a_(n-1))
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Q: Can I use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative common ratio?
A: No, you cannot use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a zero common ratio?
A: To find the common ratio of a geometric sequence with a zero common ratio, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero common ratio?
A: No, you cannot use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a negative first term and a negative common ratio?
A: To find the common ratio of a geometric sequence with a negative first term and a negative common ratio, you can use the formula:
r = (-a_n / a_(n-1))
where r is the common ratio, a_n is the nth term, and a_(n-1) is the (n-1)th term.
Q: Can I use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term and a negative common ratio?
A: No, you cannot use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term and a negative common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a zero first term and a zero common ratio?
A: To find the common ratio of a geometric sequence with a zero first term and a zero common ratio, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero first term and a zero common ratio?
A: No, you cannot use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero first term and a zero common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, and a zero first term?
A: To find the common ratio of a geometric sequence with a negative first term, a negative common ratio, and a zero first term, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, and a zero first term?
A: No, you cannot use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, and a zero first term.
Q: What if I want to find the common ratio of a geometric sequence with a zero first term, a zero common ratio, and a negative first term?
A: To find the common ratio of a geometric sequence with a zero first term, a zero common ratio, and a negative first term, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero first term, a zero common ratio, and a negative first term?
A: No, you cannot use the formula for the nth term of a geometric sequence to find the common ratio of a geometric sequence with a zero first term, a zero common ratio, and a negative first term.
Q: What if I want to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, a zero first term, and a zero common ratio?
A: To find the common ratio of a geometric sequence with a negative first term, a negative common ratio, a zero first term, and a zero common ratio, you cannot use the formula r = (a_n / a_(n-1)).
Q: Can I use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, a zero first term, and a zero common ratio?
A: No, you cannot use the formula for the sum of a geometric sequence to find the common ratio of a geometric sequence with a negative first term, a negative common ratio, a zero first term, and a zero common ratio.
Q: What if I want to find the common ratio of a geometric sequence with a zero first term, a zero common ratio, a negative first term, and a negative common ratio?
A: To find the common ratio of a geometric sequence with a zero first term, a zero common ratio, a negative first term, and a negative common ratio, you cannot use the formula r = (a_n / a_(n-1)).
**Q: Can