Finding the nth Term of a Geometric Sequence

Oct 13, 2025 by talynt.org 122 views

Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding how to find the nth term is crucial for solving various problems. In this article, we will explore the formula for finding the nth term of a geometric sequence and apply it to a given 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 form of a geometric sequence is:

a, ar, ar^2, ar^3, \ldots

where $a$ is the first term and $r$ is the common ratio.

Finding the nth Term of a Geometric Sequence

The formula for finding the nth term of a geometric sequence is given by:

a_n = a \cdot r^{n-1}

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

Example: Finding the nth Term of a Geometric Sequence

Let's consider the geometric sequence:

\frac{1}{6}, 1, 6, 36

We need to find the formula for the nth term of this sequence. To do this, we need to identify the first term and the common ratio.

Identifying the First Term and Common Ratio

The first term of the sequence is $\frac{1}{6}$ . To find the common ratio, we can divide any term by the previous term. Let's divide the second term by the first term:

\frac{1}{6} \div \frac{1}{6} = 1

This tells us that the common ratio is $1$ . However, we can also divide the third term by the second term:

6 \div 1 = 6

This tells us that the common ratio is $6$ . Since the common ratio is not constant, we need to re-examine the sequence.

Re-examining the Sequence

Let's re-examine the sequence:

\frac{1}{6}, 1, 6, 36

We can see that each term is obtained by multiplying the previous term by $6$ . Therefore, the common ratio is $6$ .

Finding the Formula for the nth Term

Now that we have identified the first term and the common ratio, we can find the formula for the nth term:

a_n = \frac{1}{6} \cdot 6^{n-1}

This formula tells us that the nth term of the sequence is equal to $\frac{1}{6}$ multiplied by $6$ raised to the power of $n-1$ .

Conclusion

In this article, we have explored the formula for finding the nth term of a geometric sequence. We have applied this formula to a given sequence and found the correct formula for the nth term. The formula for the nth term of a geometric sequence is given by:

a_n = a \cdot r^{n-1}

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

Discussion

Which formula can be used to find the nth term of the geometric sequence below?

\frac{1}{6}, 1, 6, 36

A. $a_n = 6^{n-1}$

B. $a_n = \frac{1}{6} \left(\frac{1}{6}\right)^n$

C. $a_n = \frac{1}{6} \cdot 6^{n-1}$

The correct answer is C. $a_n = \frac{1}{6} \cdot 6^{n-1}$ .

Answer Key

A. Incorrect

B. Incorrect

C. Correct

Final Thoughts

In conclusion, finding the nth term of a geometric sequence is a crucial concept in mathematics. By understanding the formula for the nth term, we can solve various problems and apply it to real-world scenarios. The formula for the nth term of a geometric sequence is given by:

a_n = a \cdot r^{n-1}

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

Introduction

In our previous article, we explored the formula for finding the nth term of a geometric sequence. In this article, we will answer some frequently asked questions about geometric sequences and provide additional examples to help solidify your understanding.

Q: What is a geometric sequence?

A: 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.

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

A: To find the common ratio, you can divide any term by the previous term. For example, if the sequence is 2, 6, 18, 54, you can divide the second term by the first term to get 6 ÷ 2 = 3. This tells us that the common ratio is 3.

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

A: The formula for finding the nth term of a geometric sequence is given by:

a_n = a \cdot r^{n-1}

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

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

A: To find the first term, you can look at the sequence and identify the first number. For example, if the sequence is 3, 9, 27, 81, the first term is 3.

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

A: 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. An arithmetic sequence, on the other hand, is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

Q: Can you provide an example of a geometric sequence?

A: Yes, here is an example of a geometric sequence:

2, 6, 18, 54

This sequence has a common ratio of 3, and the nth term can be found using the formula:

a_n = 2 \cdot 3^{n-1}

Q: Can you provide an example of a geometric sequence with a common ratio of -2?

A: Yes, here is an example of a geometric sequence with a common ratio of -2:

-4, 8, -16, 32

This sequence has a common ratio of -2, and the nth term can be found using the formula:

a_n = -4 \cdot (-2)^{n-1}

Q: Can you provide an example of a geometric sequence with a common ratio of 0?

A: No, a geometric sequence cannot have a common ratio of 0. If the common ratio is 0, the sequence would be a constant sequence, and it would not be a geometric sequence.

Q: Can you provide an example of a geometric sequence with a common ratio of 1?

A: Yes, here is an example of a geometric sequence with a common ratio of 1:

2, 2, 2, 2

This sequence has a common ratio of 1, and the nth term can be found using the formula:

a_n = 2

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences and provided additional examples to help solidify your understanding. We have also discussed the formula for finding the nth term of a geometric sequence and provided examples of geometric sequences with different common ratios.

Final Thoughts

Geometric sequences are an important concept in mathematics, and understanding how to find the nth term is crucial for solving various problems. By following the formula and examples provided in this article, you should be able to find the nth term of any geometric sequence.

Additional Resources

If you are looking for additional resources to help you understand geometric sequences, here are a few suggestions:

Khan Academy: Geometric Sequences
Mathway: Geometric Sequences
Wolfram Alpha: Geometric Sequences

Answer Key

Q: What is a geometric sequence? A: 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.

Q: How do I find the common ratio of a geometric sequence? A: To find the common ratio, you can divide any term by the previous term.

Q: What is the formula for finding the nth term of a geometric sequence? A: The formula for finding the nth term of a geometric sequence is given by:

a_n = a \cdot r^{n-1}

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

Q: How do I find the first term of a geometric sequence? A: To find the first term, you can look at the sequence and identify the first number.

Q: What is the difference between a geometric sequence and an arithmetic sequence? A: 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. An arithmetic sequence, on the other hand, is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

Q: Can you provide an example of a geometric sequence? A: Yes, here is an example of a geometric sequence:

2, 6, 18, 54

Q: Can you provide an example of a geometric sequence with a common ratio of -2? A: Yes, here is an example of a geometric sequence with a common ratio of -2:

-4, 8, -16, 32

Q: Can you provide an example of a geometric sequence with a common ratio of 0? A: No, a geometric sequence cannot have a common ratio of 0.

Q: Can you provide an example of a geometric sequence with a common ratio of 1? A: Yes, here is an example of a geometric sequence with a common ratio of 1:

2, 2, 2, 2