First 6 Terms of the Sequence

Oct 13, 2025 by windailys.com 48 views

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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 the third and fifth terms, and then use this information to determine the first 6 terms of the sequence.

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.

Finding the Common Ratio

To find the common ratio, we can use the given information about the third and fifth terms. Let's denote the third term as a_3 and the fifth term as a_5. We are given that a_3 = 32 and a_5 = 512.

Using the formula for a geometric sequence, we can write:

a_3 = a_1 * r^(3-1) = a_1 * r^2 a_5 = a_1 * r^(5-1) = a_1 * r^4

We can divide the two equations to eliminate a_1:

a_5 / a_3 = (a_1 * r^4) / (a_1 * r^2) a_5 / a_3 = r^2

Now, we can substitute the given values for a_3 and a_5:

512 / 32 = r^2 16 = r^2

To find the common ratio, we take the square root of both sides:

r = ±√16 r = ±4

Since the common ratio must be non-zero, we can choose either the positive or negative value. However, in this case, we will choose the positive value:

r = 4

First 6 Terms of the Sequence

Now that we have found the common ratio, we can use it to determine the first 6 terms of the sequence. We will start with the first term, a_1, and then use the formula for a geometric sequence to find each subsequent term.

a_1 = a_3 / r^2 a_1 = 32 / (4^2) a_1 = 32 / 16 a_1 = 2

Now, we can use the formula for a geometric sequence to find each subsequent term:

a_2 = a_1 * r a_2 = 2 * 4 a_2 = 8

a_3 = a_2 * r a_3 = 8 * 4 a_3 = 32

a_4 = a_3 * r a_4 = 32 * 4 a_4 = 128

a_5 = a_4 * r a_5 = 128 * 4 a_5 = 512

a_6 = a_5 * r a_6 = 512 * 4 a_6 = 2048

The first 6 terms of the sequence are:

2, 8, 32, 128, 512, 2048

Conclusion

In this article, we have explored how to find the common ratio of a geometric sequence given the third and fifth terms, and then used this information to determine the first 6 terms of the sequence. We have seen that the common ratio is 4, and the first 6 terms of the sequence are 2, 8, 32, 128, 512, and 2048. This demonstrates the power of geometric sequences in modeling real-world phenomena and provides a useful tool for solving problems in mathematics and other fields.

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Introduction

In our previous article, we explored how to find the common ratio of a geometric sequence given the third and fifth terms, and then used this information to determine the first 6 terms of the sequence. In this article, we will answer some frequently asked questions about geometric sequences and provide additional information to help you better understand this topic.

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: 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_m)^(1/(n-m))

where a_n is the nth term, a_m is the mth term, and n and m are the term numbers.

Alternatively, you can use the formula:

r = (a_n / a_(n-1))

where a_n is the nth term and a_(n-1) is the (n-1)th term.

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

A: To find the first term, you can use the formula:

a_1 = a_n / (r^(n-1))

where a_n is the nth term, r is the common ratio, and n is the term number.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is:

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: 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 an arithmetic 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 = 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: 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 = a_1 / (1 - r)

where S is the sum of the infinite sequence, a_1 is the first term, and r is the common ratio.

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences and provided additional information to help you better understand this topic. We have seen that geometric sequences are a powerful tool for modeling real-world phenomena and provide a useful tool for solving problems in mathematics and other fields.

Additional Resources

Final Thoughts

Geometric sequences are a fundamental concept in mathematics and have many real-world applications. By understanding how to find the common ratio and the first term of a geometric sequence, you can use this information to solve problems in mathematics and other fields. We hope that this article has provided you with a better understanding of geometric sequences and has inspired you to learn more about this topic.