The First Four Terms of the Geometric Sequence

Oct 13, 2025 by miodeya.com 87 views

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