Given The Following Geometric Sequence, Find The $12^{\text{th}}$ Term: { 2 , − 4 , 8 , … } \{2, -4, 8, \ldots\} { 2 , − 4 , 8 , … } .A. 4096 B. 2048 C. − 2048 -2048 − 2048 D. − 4096 -4096 − 4096

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. The general formula for a geometric sequence is given by: $a_n = a_1 \cdot r^{(n-1)}$, where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Identifying the Common Ratio

In the given geometric sequence $\{2, -4, 8, \ldots\}$, we can find the common ratio by dividing any term by its previous term. For example, we can divide the second term by the first term: $\frac{-4}{2} = -2$. This means that the common ratio is $-2$.

Finding the 12th Term

Now that we have the common ratio, we can use the formula for a geometric sequence to find the 12th term. We know that the first term $a_1$ is $2$, the common ratio $r$ is $-2$, and we want to find the 12th term, so $n = 12$. Plugging these values into the formula, we get: $a_{12} = 2 \cdot (-2)^{(12-1)} = 2 \cdot (-2)^{11}$.

Simplifying the Expression

To simplify the expression, we can use the property of exponents that states: $a^m \cdot a^n = a^{m+n}$. Applying this property to the expression, we get: $2 \cdot (-2)^{11} = 2 \cdot (-2)^{10} \cdot (-2) = 2 \cdot 1024 \cdot (-2) = -2048$.

Conclusion

Therefore, the 12th term of the given geometric sequence is $-2048$.

Common Mistakes to Avoid

When finding the nth term of a geometric sequence, it's easy to make mistakes. Here are a few common mistakes to avoid:

• Incorrect common ratio: Make sure to find the correct common ratio by dividing any term by its previous term.
• Incorrect exponent: Make sure to use the correct exponent when plugging the values into the formula.
• Incorrect calculation: Double-check your calculation to make sure you're getting the correct answer.

Real-World Applications

Geometric sequences have many real-world applications, including:

• Finance: Geometric sequences can be used to model population growth, compound interest, and inflation.
• Science: Geometric sequences can be used to model the growth of populations, the spread of diseases, and the decay of radioactive materials.
• Engineering: Geometric sequences can be used to model the behavior of electrical circuits, mechanical systems, and other complex systems.

Practice Problems

Here are a few practice problems to help you practice finding the nth term of a geometric sequence:

• Problem 1: Find the 8th term of the geometric sequence $\{3, 6, 12, \ldots\}$.
• Problem 2: Find the 12th term of the geometric sequence $\{4, -8, 16, \ldots\}$.
• Problem 3: Find the 10th term of the geometric sequence $\{2, 4, 8, \ldots\}$.

Conclusion

In conclusion, finding the nth term of a geometric sequence is a straightforward process that involves using the formula for a geometric sequence and plugging in the values. By following the steps outlined in this article, you should be able to find the nth term of any geometric sequence. Remember to avoid common mistakes and to practice, practice, practice!

Frequently Asked Questions

Here are some frequently asked questions about geometric sequences, along with their answers.

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, divide any term by its previous term. For example, if the sequence is $\{2, 4, 8, \ldots\}$, the common ratio is $\frac{4}{2} = 2$.

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

A: To find the nth term, 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: $a_n = a_1 \cdot r^{(n-1)}$, 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: How do I simplify the expression for the nth term?

A: To simplify the expression, use the property of exponents that states: $a^m \cdot a^n = a^{m+n}$. For example, if the expression is $2 \cdot (-2)^{11}$, you can simplify it to $2 \cdot (-2)^{10} \cdot (-2) = 2 \cdot 1024 \cdot (-2) = -2048$.

Q: What are some common mistakes to avoid when finding the nth term of a geometric sequence?

A: Some common mistakes to avoid include:

• Incorrect common ratio: Make sure to find the correct common ratio by dividing any term by its previous term.
• Incorrect exponent: Make sure to use the correct exponent when plugging the values into the formula.
• Incorrect calculation: Double-check your calculation to make sure you're getting the correct answer.

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

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

• Finance: Geometric sequences can be used to model population growth, compound interest, and inflation.
• Science: Geometric sequences can be used to model the growth of populations, the spread of diseases, and the decay of radioactive materials.
• Engineering: Geometric sequences can be used to model the behavior of electrical circuits, mechanical systems, and other complex systems.

Q: How can I practice finding the nth term of a geometric sequence?

A: You can practice finding the nth term of a geometric sequence by working through practice problems, such as:

• Problem 1: Find the 8th term of the geometric sequence $\{3, 6, 12, \ldots\}$.
• Problem 2: Find the 12th term of the geometric sequence $\{4, -8, 16, \ldots\}$.
• Problem 3: Find the 10th term of the geometric sequence $\{2, 4, 8, \ldots\}$.

Conclusion

In conclusion, geometric sequences are a fundamental concept in mathematics, and understanding how to find the nth term is crucial for many real-world applications. By following the steps outlined in this article, you should be able to find the nth term of any geometric sequence. Remember to avoid common mistakes and to practice, practice, practice!

Additional Resources

If you're looking for additional resources to help you learn more about geometric sequences, here are a few suggestions:

• Online tutorials: Websites such as Khan Academy, Mathway, and Wolfram Alpha offer interactive tutorials and practice problems to help you learn more about geometric sequences.
• Textbooks: There are many textbooks available that cover geometric sequences in detail, including "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak.
• Online communities: Join online communities such as Reddit's r/learnmath and r/math to connect with other math enthusiasts and get help with any questions you may have.

Final Tips

Here are a few final tips to help you succeed in finding the nth term of a geometric sequence:

• Practice regularly: The more you practice, the more comfortable you'll become with finding the nth term of a geometric sequence.
• Use online resources: Take advantage of online resources such as tutorials, practice problems, and online communities to help you learn more about geometric sequences.
• Seek help when needed: Don't be afraid to ask for help if you're struggling with a particular problem or concept.