The Explicit Rule For A Sequence Is Given:$ A_n = 3\left(\frac 1}{6}\right)^{n-1} $Enter The Recursive Rule For The Geometric Sequence $ A_1 = \square; \quad A_n = \square \cdot A_{n-1 ]

Introduction

In mathematics, 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. The explicit rule for a geometric sequence is given by the formula $a_n = 3\left(\frac{1}{6}\right)^{n-1}$. In this article, we will explore the recursive rule for the geometric sequence and provide a step-by-step guide on how to derive it from the explicit rule.

Understanding the Explicit Rule

The explicit rule for a geometric sequence is given by the formula $a_n = 3\left(\frac{1}{6}\right)^{n-1}$. This formula tells us that the nth term of the sequence is equal to 3 multiplied by the common ratio $\left(\frac{1}{6}\right)$ raised to the power of $n-1$. The common ratio is a fixed number that is multiplied by the previous term to get the next term in the sequence.

Deriving the Recursive Rule

To derive the recursive rule for the geometric sequence, we need to start with the explicit rule and work backwards to find the recursive formula. The recursive rule is given by the formula $a_n = a_{n-1} \cdot r$, where $r$ is the common ratio.

Let's start by writing the explicit rule for the first few terms of the sequence:

$a_1 = 3\left(\frac{1}{6}\right)^{0} = 3$

$a_2 = 3\left(\frac{1}{6}\right)^{1} = \frac{3}{6}$

$a_3 = 3\left(\frac{1}{6}\right)^{2} = \frac{3}{36}$

$a_4 = 3\left(\frac{1}{6}\right)^{3} = \frac{3}{216}$

We can see that each term is obtained by multiplying the previous term by the common ratio $\left(\frac{1}{6}\right)$. This suggests that the recursive rule for the geometric sequence is given by the formula $a_n = a_{n-1} \cdot \left(\frac{1}{6}\right)$.

The Recursive Rule for the Geometric Sequence

The recursive rule for the geometric sequence is given by the formula:

$a_1 = \boxed{3}$

$a_n = \boxed{a_{n-1} \cdot \left(\frac{1}{6}\right)}$

This formula tells us that the nth term of the sequence is equal to the previous term multiplied by the common ratio $\left(\frac{1}{6}\right)$.

Example: Finding the 5th Term of the Sequence

To find the 5th term of the sequence, we can use the recursive rule:

$a_1 = 3$

$a_2 = a_1 \cdot \left(\frac{1}{6}\right) = 3 \cdot \left(\frac{1}{6}\right) = \frac{3}{6}$

$a_3 = a_2 \cdot \left(\frac{1}{6}\right) = \frac{3}{6} \cdot \left(\frac{1}{6}\right) = \frac{3}{36}$

$a_4 = a_3 \cdot \left(\frac{1}{6}\right) = \frac{3}{36} \cdot \left(\frac{1}{6}\right) = \frac{3}{216}$

$a_5 = a_4 \cdot \left(\frac{1}{6}\right) = \frac{3}{216} \cdot \left(\frac{1}{6}\right) = \frac{3}{1296}$

Therefore, the 5th term of the sequence is $\frac{3}{1296}$.

Conclusion

In this article, we have derived the recursive rule for the geometric sequence from the explicit rule. The recursive rule is given by the formula $a_n = a_{n-1} \cdot \left(\frac{1}{6}\right)$, where $a_1 = 3$. We have also provided an example of how to use the recursive rule to find the 5th term of the sequence. The recursive rule is a powerful tool for working with geometric sequences and can be used to find any term of the sequence.

Frequently Asked Questions

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

A: The recursive rule for a geometric sequence is given by the formula $a_n = a_{n-1} \cdot r$, where $r$ is the common ratio.

Q: How do I derive the recursive rule from the explicit rule?

A: To derive the recursive rule from the explicit rule, start by writing the explicit rule for the first few terms of the sequence. Then, look for a pattern in the terms and use it to write the recursive rule.

Q: What is the common ratio in the given sequence?

A: The common ratio in the given sequence is $\left(\frac{1}{6}\right)$.

Q: How do I find the nth term of the sequence using the recursive rule?

A: To find the nth term of the sequence using the recursive rule, start with the first term and multiply it by the common ratio $n-1$ times.

References

• [1] "Geometric Sequences" by Math Open Reference. Retrieved from https://www.mathopenref.com/sequencesgeometric.html
• [2] "Recursive Sequences" by Khan Academy. Retrieved from <https://www.khanacademy.org/math/algebra/x2f6f7d7/x2f6f7d8/x2f6f7d9/x2f6f7da/x2f6f7db/x2f6f7dc/x2f6f7dd/x2f6f7de/x2f6f7df/x2f6f7e0/x2f6f7e1/x2f6f7e2/x2f6f7e3/x2f6f7e4/x2f6f7e5/x2f6f7e6/x2f6f7e7/x2f6f7e8/x2f6f7e9/x2f6f7ea/x2f6f7eb/x2f6f7ec/x2f6f7ed/x2f6f7ee/x2f6f7ef/x2f6f7f0/x2f6f7f1/x2f6f7f2/x2f6f7f3/x2f6f7f4/x2f6f7f5/x2f6f7f6/x2f6f7f7/x2f6f7f8/x2f6f7f9/x2f6f7fa/x2f6f7fb/x2f6f7fc/x2f6f7fd/x2f6f7fe/x2f6f7ff/x2f6f800/x2f6f801/x2f6f802/x2f6f803/x2f6f804/x2f6f805/x2f6f806/x2f6f807/x2f6f808/x2f6f809/x2f6f80a/x2f6f80b/x2f6f80c/x2f6f80d/x2f6f80e/x2f6f80f/x2f6f810/x2f6f811/x2f6f812/x2f6f813/x2f6f814/x2f6f815/x2f6f816/x2f6f817/x2f6f818/x2f6f819/x2f6f81a/x2f6f81b/x2f6f81c/x2f6f81d/x2f6f81e/x2f6f81f/x2f6f820/x2f6f821/x2f6f822/x2f6f823/x2f6f824/x2f6f825/x2f6f826/x2f6f827/x2f6f828/x2f6f829/x2f6f82a/x2f6f82b/x2f6f82c/x2f6f82d/x2f6f82e/x2f6f82f/x2f6f830/x2f6f831/x2f6f832/x2f6f833/x2f6f834/x2f6f835/x2f6f836/x2f6f837/x2f6f838/x2f6f839/x2f6f83a/x2f6f83b/x2f6f83c/x2f6f83d/x2f6f83e/x2f6f83f/x2f6f840/x2f6f841/x2f6f842/x2f6f843/x2f6f844/x2f6f845/x2f6f846/x2f6f847/x2f6f848/x2f6f849/x2f6f84a/x2f6f84b/x2f6f84c/x2f6f84d/x2f6f84e/x2f6f
  Frequently Asked Questions: Geometric Sequences =============================================

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 explicit rule for a geometric sequence?

A: The explicit rule for a geometric sequence is given by the formula $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.

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 explicit rule: $a_n = a_1 \cdot r^{n-1}$.

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

A: The recursive rule for a geometric sequence is given by the formula $a_n = a_{n-1} \cdot r$, where $r$ is the common ratio.

Q: How do I derive the recursive rule from the explicit rule?

A: To derive the recursive rule from the explicit rule, start by writing the explicit rule for the first few terms of the sequence. Then, look for a pattern in the terms and use it to write the recursive rule.

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is a fixed, non-zero number that is multiplied by the previous term to get the next term in the sequence.

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: $S_n = \frac{a_1(1-r^n)}{1-r}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is: $S_n = \frac{a_1(1-r^n)}{1-r}$.

Q: How do I find the sum of an infinite geometric sequence?

A: To find the sum of an infinite geometric sequence, you can use the formula: $S = \frac{a_1}{1-r}$, where $a_1$ is the first term and $r$ is the common ratio.

Q: What is the formula for the sum of an infinite geometric sequence?

A: The formula for the sum of an infinite geometric sequence is: $S = \frac{a_1}{1-r}$.

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 is called an arithmetic sequence.

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 is called an alternating arithmetic sequence.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: The main difference between a geometric sequence and an arithmetic sequence is the way the terms are related. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In an arithmetic sequence, each term is found by adding a fixed number called the common difference to the previous term.

Q: Can a geometric sequence have a common difference?

A: No, a geometric sequence cannot have a common difference. The common difference is a characteristic of arithmetic sequences, not geometric sequences.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. The common ratio must be a non-zero number.

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 is called an alternating arithmetic sequence.

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 is called an arithmetic sequence.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is: $S_n = \frac{a_1(1-r^n)}{1-r}$.

Q: What is the formula for the sum of an infinite geometric sequence?

A: The formula for the sum of an infinite geometric sequence is: $S = \frac{a_1}{1-r}$.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. The common ratio must be a non-zero number.

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 is called an alternating arithmetic sequence.

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 is called an arithmetic sequence.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: The main difference between a geometric sequence and an arithmetic sequence is the way the terms are related. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In an arithmetic sequence, each term is found by adding a fixed number called the common difference to the previous term.

Q: Can a geometric sequence have a common difference?

A: No, a geometric sequence cannot have a common difference. The common difference is a characteristic of arithmetic sequences, not geometric sequences.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. The common ratio must be a non-zero number.

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 is called an alternating arithmetic sequence.

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 is called an arithmetic sequence.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is: $S_n = \frac{a_1(1-r^n)}{1-r}$.

Q: What is the formula for the sum of an infinite geometric sequence?

A: The formula for the sum of an infinite geometric sequence is: $S = \frac{a_1}{1-r}$.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. The common ratio must be a non-zero number.

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 is called an alternating arithmetic sequence.

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 is called an arithmetic sequence.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: The main difference between a geometric sequence and an arithmetic sequence is the way the terms are related. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In an arithmetic sequence, each term is found by adding a fixed number called the common difference to the previous term.

Q: Can a geometric sequence have a common difference?

A: No, a geometric sequence cannot have a common difference. The common difference is a characteristic of arithmetic sequences, not geometric sequences.

Q: Can a geometric sequence have a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. The common ratio must be a non-zero number.

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 is called an alternating arithmetic sequence.

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