Finding the Recursive Rule for a Geometric Sequence

Oct 13, 2025 by wbrum.org 107 views

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 a recursive rule for a geometric sequence and provide a step-by-step guide on how to find it.

What is a Recursive Rule?

A recursive rule is a formula that defines each term of a sequence as a function of the preceding term(s). In other words, it is a rule that tells us how to get from one term to the next. For a geometric sequence, the recursive rule is typically of the form:

a_n = r \cdot a_{n-1}

where $a_n$ is the $n$ th term of the sequence, $r$ is the common ratio, and $a_{n-1}$ is the $(n-1)$ th term.

Finding the Recursive Rule for a Geometric Sequence

To find the recursive rule for a geometric sequence, we need to identify the common ratio $r$ . Here's a step-by-step guide:

Step 1: Identify the First Term

The first term of the sequence is given as $a_1 = 6$ .

Step 2: Identify the Second Term

The second term of the sequence is given as $a_2 = -18$ .

Step 3: Find the Common Ratio

To find the common ratio $r$ , we can divide the second term by the first term:

r = \frac{a_2}{a_1} = \frac{-18}{6} = -3

Step 4: Write the Recursive Rule

Now that we have found the common ratio $r$ , we can write the recursive rule for the geometric sequence:

a_n = -3 \cdot a_{n-1}

Example: Recursive Rule for the Geometric Sequence

Let's consider the geometric sequence:

6, -18, 54, -162, \ldots

To find the recursive rule for this sequence, we can follow the steps outlined above:

Step 1: Identify the First Term

The first term of the sequence is given as $a_1 = 6$ .

Step 2: Identify the Second Term

The second term of the sequence is given as $a_2 = -18$ .

Step 3: Find the Common Ratio

To find the common ratio $r$ , we can divide the second term by the first term:

r = \frac{a_2}{a_1} = \frac{-18}{6} = -3

Step 4: Write the Recursive Rule

Now that we have found the common ratio $r$ , we can write the recursive rule for the geometric sequence:

a_n = -3 \cdot a_{n-1}

Conclusion

In this article, we have explored the concept of a recursive rule for a geometric sequence. We have provided a step-by-step guide on how to find the recursive rule for a given geometric sequence. By following these steps, you can easily find the recursive rule for any geometric sequence.

Frequently Asked Questions

Q: What is a recursive rule for a geometric sequence?

A: A recursive rule for a geometric sequence is a formula that defines each term of the sequence as a function of the preceding term(s).

Q: How do I find the recursive rule for a geometric sequence?

A: To find the recursive rule for a geometric sequence, you need to identify the common ratio $r$ and then write the recursive rule in the form $a_n = r \cdot a_{n-1}$ .

Q: What is the common ratio $r$ ?

A: The common ratio $r$ is a fixed, non-zero number that is used to multiply the previous term to get the next term in the sequence.

References

[1] "Geometric Sequences" by Math Open Reference. Retrieved from https://www.mathopenref.com/sequencesgeometric.html
[2] "Recursive Rules" by Khan Academy. Retrieved from https://www.khanacademy.org/math/sequences-series-and-testing/sequences-and-series/recursive-rules/v/recursive-rules

Glossary

Geometric sequence: 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.
Recursive rule: A formula that defines each term of a sequence as a function of the preceding term(s).
Common ratio: A fixed, non-zero number that is used to multiply the previous term to get the next term in the sequence.
Geometric Sequence Recursive Rule Q&A =====================================

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 a recursive rule for a geometric sequence?

A: A recursive rule for a geometric sequence is a formula that defines each term of the sequence as a function of the preceding term(s). It is typically of the form $a_n = r \cdot a_{n-1}$ , where $a_n$ is the $n$ th term of the sequence, $r$ is the common ratio, and $a_{n-1}$ is the $(n-1)$ th term.

Q: How do I find the recursive rule for a geometric sequence?

A: To find the recursive rule for a geometric sequence, you need to identify the common ratio $r$ and then write the recursive rule in the form $a_n = r \cdot a_{n-1}$ . You can find the common ratio by dividing the second term by the first term.

Q: What is the common ratio $r$ ?

A: The common ratio $r$ is a fixed, non-zero number that is used to multiply the previous term to get the next term in the sequence.

Q: How do I find the common ratio $r$ ?

A: To find the common ratio $r$ , you can divide the second term by the first term. For example, if the first term is $a_1 = 6$ and the second term is $a_2 = -18$ , then the common ratio $r$ is $r = \frac{a_2}{a_1} = \frac{-18}{6} = -3$ .

Q: What is the formula for the $n$ th term of a geometric sequence?

A: The formula for the $n$ th term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$ , where $a_n$ is the $n$ th 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 use the recursive rule to find the $n$ th term of a geometric sequence?

A: To use the recursive rule to find the $n$ th term of a geometric sequence, you can start with the first term $a_1$ and then repeatedly apply the recursive rule to find the next term. For example, if the recursive rule is $a_n = -3 \cdot a_{n-1}$ , then you can find the second term by multiplying the first term by $-3$ , the third term by multiplying the second term by $-3$ , and so on.

Q: What is the difference between a recursive rule and a formula for the $n$ th term?

A: A recursive rule is a formula that defines each term of a sequence as a function of the preceding term(s), while a formula for the $n$ th term is a formula that gives the value of the $n$ th term directly. The recursive rule is useful for finding the terms of a sequence one at a time, while the formula for the $n$ th term is useful for finding the value of a specific term.

Q: Can I use the recursive rule to find the value of a specific term of a geometric sequence?

A: Yes, you can use the recursive rule to find the value of a specific term of a geometric sequence. To do this, you can start with the first term and then repeatedly apply the recursive rule to find the next term until you reach the term you want to find.

Q: How do I know if a sequence is a geometric sequence?

A: To determine if a sequence is a geometric sequence, you can check if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. If this is the case, then the sequence is a geometric sequence.

Q: What are some common applications of geometric sequences?

A: Geometric sequences have many common applications in mathematics, science, and engineering. Some examples include:

Modeling population growth or decline
Describing the behavior of electrical circuits
Analyzing the motion of objects under constant acceleration
Calculating the value of investments or annuities

Q: Can I use the recursive rule to find the sum of a geometric sequence?

A: Yes, you can use the recursive rule to find the sum of a geometric sequence. To do this, you can use the formula for the sum of a geometric series, which is $S_n = \frac{a_1(1-r^n)}{1-r}$ , 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: What are some common mistakes to avoid when working with geometric sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

Assuming that the sequence is arithmetic instead of geometric
Failing to check if the sequence is a geometric sequence before applying the recursive rule
Making errors when calculating the common ratio or the recursive rule
Failing to consider the possibility of a zero or negative common ratio.