How to Find the Fifth Term of a Geometric Sequence

Oct 13, 2025 by miodeya.com 16 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 discuss how to find the fifth term of a geometric sequence given the first term and the common ratio.

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:

an = a1 * r^(n-1)

where:

an is the nth term of the sequence
a1 is the first term of the sequence
r is the common ratio
n is the term number

Example: Finding the Fifth Term of a Geometric Sequence

In this example, we are given the first term (a1) and the common ratio (r). We need to find the fifth term (a5) of the geometric sequence.

Given:

a1 = 2
r = 3
n = 5

We can use the formula for a geometric sequence to find the fifth term:

a5 = a1 * r^(n-1) = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162

How to Find the Fifth Term of a Geometric Sequence

To find the fifth term of a geometric sequence, you can use the formula:

an = a1 * r^(n-1)

where:

an is the nth term of the sequence
a1 is the first term of the sequence
r is the common ratio
n is the term number

You can plug in the values of a1, r, and n into the formula to find the fifth term.

Step-by-Step Solution

Here's a step-by-step solution to find the fifth term of a geometric sequence:

Write down the formula for a geometric sequence: an = a1 * r^(n-1)
Plug in the values of a1, r, and n into the formula: a5 = 2 * 3^(5-1)
Simplify the expression: a5 = 2 * 3^4
Calculate the value of 3^4: 3^4 = 81
Multiply 2 by 81: a5 = 2 * 81 = 162

Conclusion

In this article, we discussed how to find the fifth term of a geometric sequence given the first term and the common ratio. We used the formula for a geometric sequence to find the fifth term and provided a step-by-step solution to illustrate the process.

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 fifth term of a geometric sequence?

A: To find the fifth term of a geometric sequence, you can use the formula: an = a1 * r^(n-1), where a1 is the first term, r is the common ratio, and n is the term number.

Q: What is the common ratio?

A: The common ratio is a fixed, non-zero number that is used to find each term of a geometric sequence.

Q: How do I calculate the common ratio?

A: The common ratio can be calculated by dividing any term by its previous term.

Additional Resources

For more information on geometric sequences, you can check out the following resources:

References

Geometric Sequence
Geometric Sequence Formula
Geometric Sequence Examples
Geometric Sequence Q&A =========================

Introduction

In our previous article, we discussed how to find the fifth term of a geometric sequence given the first term and the common ratio. In this article, we will answer some frequently asked questions about geometric sequences.

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 first term of a geometric sequence?

A: To find the first term of a geometric sequence, you can use the formula: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number. You can rearrange the formula to solve for a1: a1 = an / (r^(n-1)).

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

A: To find the common ratio of a geometric sequence, you can divide any term by its previous term. For example, if you know the first two terms of the sequence, you can find the common ratio by dividing the second term by the first term: r = a2 / a1.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

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 = a1 * (1 - r^n) / (1 - r), where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

Q: What is the difference between a geometric sequence and an arithmetic 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. An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

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: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

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

A: The formula for the sum of a geometric sequence is: S = a1 * (1 - r^n) / (1 - r), where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

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 = a1 / (1 - r), where S is the sum, a1 is the first term, and r is the common ratio.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences. We hope that this article has been helpful in understanding geometric sequences and how to work with them.

Additional Resources

For more information on geometric sequences, you can check out the following resources: