Use The Recursively Defined Geometric Sequence $a_1 = \frac{5}{6}, A_n = 4a_{n-1}$ And Find The Common Ratio.A. $\frac{10}{3}$B. $-\frac{2}{3}$C. $\frac{5}{6}$D. 4

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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 general formula for a geometric sequence is given by:

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.

In this article, we will use a recursively defined geometric sequence to find the common ratio. The sequence is defined as:

a_1 = \frac{5}{6}, a_n = 4a_{n-1}

Our goal is to find the common ratio, r.

Understanding the Sequence

To find the common ratio, we need to understand how the sequence is defined. The sequence is recursively defined, meaning that each term is defined in terms of the previous term. In this case, the nth term is defined as 4 times the (n-1)th term.

Let's start by finding the second term of the sequence:

a_2 = 4a_1 = 4(\frac{5}{6}) = \frac{20}{6} = \frac{10}{3}

Now, let's find the third term of the sequence:

a_3 = 4a_2 = 4(\frac{10}{3}) = \frac{40}{3}

As we can see, the sequence is increasing by a factor of 4 each time.

Finding the Common Ratio

To find the common ratio, we need to find the ratio between consecutive terms. Let's find the ratio between the second and first terms:

r = \frac{a_2}{a_1} = \frac{\frac{10}{3}}{\frac{5}{6}} = \frac{10}{3} * \frac{6}{5} = \frac{60}{15} = 4

However, this is not the common ratio. The common ratio is the ratio between consecutive terms, not the ratio between the second term and the first term.

Let's try again. Let's find the ratio between the third and second terms:

r = \frac{a_3}{a_2} = \frac{\frac{40}{3}}{\frac{10}{3}} = \frac{40}{3} * \frac{3}{10} = \frac{120}{30} = 4

Again, this is not the common ratio. We need to find the ratio between consecutive terms, not the ratio between the third term and the second term.

Let's try again. Let's find the ratio between the second and first terms, and the ratio between the third and second terms:

r = \frac{a_2}{a_1} = 4 r = \frac{a_3}{a_2} = 4

As we can see, the ratio between consecutive terms is 4. Therefore, the common ratio is 4.

Conclusion

In this article, we used a recursively defined geometric sequence to find the common ratio. The sequence was defined as:

a_1 = \frac{5}{6}, a_n = 4a_{n-1}

We found the common ratio by finding the ratio between consecutive terms. The common ratio is 4.

Answer

The common ratio is 4.

Discussion

This problem is a great example of how to find the common ratio of a geometric sequence. The sequence is recursively defined, meaning that each term is defined in terms of the previous term. We found the common ratio by finding the ratio between consecutive terms.

This problem is also a great example of how to use the formula for a geometric sequence:

a_n = a_1 * r^(n-1)

We can use this formula to find the nth term of the sequence, given the first term and the common ratio.

References

  • [1] "Geometric Sequences" by Math Open Reference
  • [2] "Geometric Sequences" by Khan Academy

Related Topics

  • [1] "Arithmetic Sequences"
  • [2] "Geometric Sequences with a Common Ratio of 1"
  • [3] "Geometric Sequences with a Common Ratio of -1"

Frequently Asked Questions

  • 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 find the ratio between consecutive terms.
  • Q: What is the formula for a geometric sequence? A: The formula for a geometric sequence is given by:

a_n = a_1 * r^(n-1)

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 find the ratio between consecutive terms. You can do this by dividing each term by the previous term.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is given by:

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 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 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 obtained. In a geometric sequence, each term is obtained by multiplying the previous term by a fixed number, while in an arithmetic sequence, each term is obtained by adding a fixed number to the previous term.

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 a constant sequence, and each term is equal to the first term.

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 sequence, and each term alternates between positive and negative values.

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 = 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: 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, a_1 is the first term, r is the common ratio, and n is the term number.

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

A: The common ratio is a crucial component of a geometric sequence, as it determines the rate at which the terms increase or decrease. A common ratio of 1 indicates a constant sequence, while a common ratio greater than 1 indicates an increasing sequence, and a common ratio less than 1 indicates a decreasing sequence.

Q: Can a geometric sequence have a common ratio that is a fraction?

A: Yes, a geometric sequence can have a common ratio that is a fraction. In this case, the sequence is still geometric, and the terms are obtained by multiplying the previous term by the fraction.

Q: How do I determine if a geometric sequence is convergent or divergent?

A: To determine if a geometric sequence is convergent or divergent, you need to check if the absolute value of the common ratio is less than 1. If this is the case, then the sequence is convergent, and if the absolute value of the common ratio is greater than 1, then the sequence is divergent.

Q: What is the relationship between the common ratio and the sum of a geometric sequence?

A: The common ratio and the sum of a geometric sequence are related by 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 number of terms.

Q: Can a geometric sequence have a common ratio that is a complex number?

A: Yes, a geometric sequence can have a common ratio that is a complex number. In this case, the sequence is still geometric, and the terms are obtained by multiplying the previous term by the complex number.

Q: How do I find the product of a geometric sequence?

A: To find the product of a geometric sequence, you can use the formula:

P_n = a_1 * r^(n-1)

where P_n is the product 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 is the relationship between the common ratio and the product of a geometric sequence?

A: The common ratio and the product of a geometric sequence are related by the formula:

P_n = a_1 * r^(n-1)

where P_n is the product of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.

Q: Can a geometric sequence have a common ratio that is a matrix?

A: Yes, a geometric sequence can have a common ratio that is a matrix. In this case, the sequence is still geometric, and the terms are obtained by multiplying the previous term by the matrix.

Q: How do I find the sum of a geometric sequence with a complex common ratio?

A: To find the sum of a geometric sequence with a complex common ratio, 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 complex common ratio, and n is the number of terms.

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

A: The complex common ratio is a crucial component of a geometric sequence, as it determines the rate at which the terms increase or decrease. A complex common ratio can result in a sequence that oscillates between positive and negative values.

Q: Can a geometric sequence have a common ratio that is a vector?

A: Yes, a geometric sequence can have a common ratio that is a vector. In this case, the sequence is still geometric, and the terms are obtained by multiplying the previous term by the vector.

Q: How do I find the product of a geometric sequence with a complex common ratio?

A: To find the product of a geometric sequence with a complex common ratio, you can use the formula:

P_n = a_1 * r^(n-1)

where P_n is the product of the first n terms, a_1 is the first term, r is the complex common ratio, and n is the number of terms.

Q: What is the relationship between the complex common ratio and the product of a geometric sequence?

A: The complex common ratio and the product of a geometric sequence are related by the formula:

P_n = a_1 * r^(n-1)

where P_n is the product of the first n terms, a_1 is the first term, r is the complex common ratio, and n is the number of terms.