Find The Fourth Term Of The Geometric Sequence \[$4, -20, 100, \ldots\$\]$\[ A_4 = \\]

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 fourth term of a geometric sequence given the first three terms.

Understanding Geometric Sequences

A geometric sequence is defined as a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. The general formula for a geometric sequence is:

$$a_n = a_1 \cdot r^{(n-1)}$$

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 fourth term of a geometric sequence, we need to find the common ratio. The common ratio can be found by dividing any term by its previous term. In this case, we can divide the second term by the first term to find the common ratio.

$$r = \frac{a_2}{a_1} = \frac{-20}{4} = -5$$

Finding the Fourth Term

Now that we have found the common ratio, we can use the formula for a geometric sequence to find the fourth term.

$$a_4 = a_1 \cdot r^{(4-1)} = 4 \cdot (-5)^3 = 4 \cdot (-125) = -500$$

Conclusion

In this article, we have explored how to find the fourth term of a geometric sequence given the first three terms. We have used the formula for a geometric sequence and found the common ratio to find the fourth term. The fourth term of the geometric sequence is -500.

Example Problems

Here are a few example problems to help you practice finding the fourth term of a geometric sequence.

Example 1

Find the fourth term of the geometric sequence: $[2, 6, 18, \ldots]$

Solution

To find the fourth term, we need to find the common ratio. We can divide the second term by the first term to find the common ratio.

$$r = \frac{a_2}{a_1} = \frac{6}{2} = 3$$

Now that we have found the common ratio, we can use the formula for a geometric sequence to find the fourth term.

$$a_4 = a_1 \cdot r^{(4-1)} = 2 \cdot 3^3 = 2 \cdot 27 = 54$$

Example 2

Find the fourth term of the geometric sequence: $[3, -9, 27, \ldots]$

Solution

To find the fourth term, we need to find the common ratio. We can divide the second term by the first term to find the common ratio.

$$r = \frac{a_2}{a_1} = \frac{-9}{3} = -3$$

Now that we have found the common ratio, we can use the formula for a geometric sequence to find the fourth term.

$$a_4 = a_1 \cdot r^{(4-1)} = 3 \cdot (-3)^3 = 3 \cdot (-27) = -81$$

Tips and Tricks

Here are a few tips and tricks to help you find the fourth term of a geometric sequence.

• Make sure to find the common ratio before finding the fourth term.
• Use the formula for a geometric sequence to find the fourth term.
• Check your work by plugging in the values into the formula.

Conclusion

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 divide any term by its previous term. For example, if the first term is 4 and the second term is -20, the common ratio would be -20/4 = -5.

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:

$$a_n = a_1 \cdot r^{(n-1)}$$

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 formula for a geometric sequence?

A: The formula for a geometric sequence is:

$$a_n = a_1 \cdot r^{(n-1)}$$

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

A: To find the fourth term of a geometric sequence, you need to find the common ratio and then use the formula for a geometric sequence. For example, if the first term is 4 and the common ratio is -5, the fourth term would be:

$$a_4 = a_1 \cdot r^{(4-1)} = 4 \cdot (-5)^3 = 4 \cdot (-125) = -500$$

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

A: An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed number to the previous term. A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed number.

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

A: To determine if a sequence is geometric or arithmetic, you can look at the relationship between the terms. If the terms are increasing or decreasing by a fixed number, it is an arithmetic sequence. If the terms are increasing or decreasing by a fixed ratio, it is a geometric sequence.

Q: What are some real-world applications of geometric sequences?

A: Geometric sequences have many real-world applications, including:

• Compound interest: When money is invested at a fixed interest rate, the interest earned is a geometric sequence.
• Population growth: When a population grows at a fixed rate, the population size is a geometric sequence.
• Music: The frequencies of notes in music are a geometric sequence.

Q: How do I use geometric sequences in real-world problems?

A: To use geometric sequences in real-world problems, you need to identify the first term, the common ratio, and the term number. Then, you can use the formula for a geometric sequence to find the desired term.

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences. We have covered topics such as finding the common ratio, finding the nth term, and real-world applications of geometric sequences. We hope this article has been helpful in understanding geometric sequences and how to use them in real-world problems.