Find The Next Term In The Sequence:$\[ 128, 64, 32, \ldots \\]A. 22 B. 24 C. 30 D. 16

Feb 27, 2025 by ADMIN 91 views

**Finding the Next Term in a Geometric Sequence**

Understanding Geometric Sequences

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 next term in a geometric sequence using the given sequence: $[128, 64, 32, \ldots]$ .

Identifying the Common Ratio

To find the next term in the sequence, we need to identify the common ratio. The common ratio is the number by which each term is multiplied to get the next term. In this case, we can find the common ratio by dividing each term by the previous term.

Let's calculate the common ratio:

$64 ÷ 128 = 0.5$
$32 ÷ 64 = 0.5$

As we can see, the common ratio is $0.5$ . This means that each term in the sequence is obtained by multiplying the previous term by $0.5$ .

Finding the Next Term

Now that we have identified the common ratio, we can find the next term in the sequence. To do this, we simply multiply the last term by the common ratio.

Let's calculate the next term:

$32 × 0.5 = 16$

Therefore, the next term in the sequence is $16$ .

Conclusion

In this article, we have learned how to find the next term in a geometric sequence using the given sequence: $[128, 64, 32, \ldots]$ . We identified the common ratio by dividing each term by the previous term and found the next term by multiplying the last term by the common ratio. The next term in the sequence is $16$ .

Real-World Applications

Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate compound interest and investment returns.
Biology: Geometric sequences are used to model population growth and decline.
Computer Science: Geometric sequences are used to model the growth of algorithms and data structures.

Example Problems

Here are some example problems to help you practice finding the next term in a geometric sequence:

Find the next term in the sequence: $[2, 6, 18, \ldots]$
Find the next term in the sequence: $[3, 9, 27, \ldots]$
Find the next term in the sequence: $[4, 16, 64, \ldots]$

Solutions

Here are the solutions to the example problems:

Find the next term in the sequence: $[2, 6, 18, \ldots]$ $[2, 6, 18, \dots]$
- Common ratio: $6 ÷ 2 = 3$
- Next term: $18 × 3 = 54$
Find the next term in the sequence: $[3, 9, 27, \ldots]$ $[3, 9, 27, \dots]$
- Common ratio: $9 ÷ 3 = 3$
- Next term: $27 × 3 = 81$
Find the next term in the sequence: $[4, 16, 64, \ldots]$ $[4, 16, 64, \dots]$
- Common ratio: $16 ÷ 4 = 4$
- Next term: $64 × 4 = 256$

Tips and Tricks

Here are some tips and tricks to help you find the next term in a geometric sequence:

Identify the common ratio: The common ratio is the key to finding the next term in a geometric sequence.
Use the formula: The formula for finding the next term in a geometric sequence is: $a_n = a_{n-1} × r$ , where $a_n$ is the next term, $a_{n-1}$ is the previous term, and $r$ is the common ratio.
Practice, practice, practice: The more you practice finding the next term in a geometric sequence, the more comfortable you will become with the concept.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about geometric sequences.

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?

A: To find the common ratio, you can divide each term by the previous term. For example, if the sequence is $[2, 6, 18, \ldots]$ , you can find the common ratio by dividing each term by the previous term: $6 ÷ 2 = 3$ , $18 ÷ 6 = 3$ . Therefore, the common ratio is $3$ .

Q: How do I find the next term in a geometric sequence?

A: To find the next term in a geometric sequence, you can multiply the last term by the common ratio. For example, if the sequence is $[2, 6, 18, \ldots]$ and the common ratio is $3$ , you can find the next term by multiplying the last term by the common ratio: $18 × 3 = 54$ .

Q: What is the formula for finding the next term in a geometric sequence?

A: The formula for finding the next term in a geometric sequence is: $a_n = a_{n-1} × r$ , where $a_n$ is the next term, $a_{n-1}$ is the previous term, and $r$ is the common ratio.

Q: Can I use a geometric sequence to model real-world situations?

A: Yes, geometric sequences can be used to model real-world situations such as population growth, compound interest, and investment returns.

Q: What are some examples of geometric sequences in real life?

A: Some examples of geometric sequences in real life include:

Population growth: The population of a city can be modeled using a geometric sequence, where each term represents the population at a given time.
Compound interest: The amount of money in a savings account can be modeled using a geometric sequence, where each term represents the amount of money at a given time.
Investment returns: The returns on an investment can be modeled using a geometric sequence, where each term represents the return at a given time.

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

A: To determine if a sequence is a geometric sequence, you can check if each term is obtained by multiplying the previous term by a fixed, non-zero number. If this is the case, then the sequence is a geometric sequence.

Q: What are some common mistakes to avoid when working with geometric sequences?

A: Some common mistakes to avoid when working with geometric sequences include:

Not identifying the common ratio: Failing to identify the common ratio can lead to incorrect calculations.
Not using the correct formula: Using the wrong formula can lead to incorrect calculations.
Not checking for convergence: Failing to check for convergence can lead to incorrect conclusions.

Q: How do I practice finding the next term in a geometric sequence?

A: You can practice finding the next term in a geometric sequence by:

Using online resources: There are many online resources available that provide practice problems and exercises for finding the next term in a geometric sequence.
Working with a tutor: Working with a tutor can provide personalized instruction and practice.
Creating your own problems: Creating your own problems can help you to practice finding the next term in a geometric sequence in a more engaging and interactive way.

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world situations and can be used to find the next term in a sequence. By understanding the common ratio and using the formula, you can find the next term in a geometric sequence with ease. Remember to practice regularly and to avoid common mistakes to become proficient in finding the next term in a geometric sequence.