A Geometric Sequence Begins With \[$2, -8, 32, -128, 512, \ldots\$\].Which Option Below Represents The Formula For The Sequence?A. \[$f(n) = 4(-2)^n\$\] B. \[$f(n) = 4(-2)^{n-1}\$\] C. \[$f(n) = 2(-4)^{n-1}\$\] D.

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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 given sequence ${2, -8, 32, -128, 512, \ldots\$} is a geometric sequence, and we need to find the formula that represents this sequence.

Identifying the Common Ratio

To find the formula for the sequence, we first 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 sequence, we can see that each term is obtained by multiplying the previous term by -4.

For example, to get from the first term 2 to the second term -8, we multiply 2 by -4. Similarly, to get from the second term -8 to the third term 32, we multiply -8 by -4.

Therefore, the common ratio is -4.

Finding the Formula

Now that we have identified the common ratio, we can find the formula for the sequence. The formula for a geometric sequence is given by:

f(n)=arn−1f(n) = ar^{n-1}

where aa is the first term, rr is the common ratio, and nn is the term number.

In this case, the first term aa is 2, and the common ratio rr is -4. Therefore, the formula for the sequence is:

f(n)=2(−4)n−1f(n) = 2(-4)^{n-1}

Evaluating the Options

Now that we have found the formula for the sequence, let's evaluate the options given:

A. {f(n) = 4(-2)^n$}$

This option is incorrect because the common ratio is -4, not -2.

B. {f(n) = 4(-2)^{n-1}$}$

This option is incorrect because the first term is 2, not 4.

C. {f(n) = 2(-4)^{n-1}$}$

This option is correct because it matches the formula we found earlier.

D. f(n)=2(−4)n${f(n) = 2(-4)^n\$}

This option is incorrect because the exponent should be n−1n-1, not nn.

Conclusion

In conclusion, the formula for the given geometric sequence is {f(n) = 2(-4)^{n-1}$}$. This formula represents the sequence ${2, -8, 32, -128, 512, \ldots\$} and can be used to find any term in the sequence.

Understanding Geometric Sequences: Key Takeaways

  • 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 common ratio is the number by which each term is multiplied to get the next term.
  • The formula for a geometric sequence is given by {f(n) = ar^{n-1}$}$, where aa is the first term, rr is the common ratio, and nn is the term number.
  • To find the formula for a geometric sequence, we need to identify the common ratio and the first term.
  • The formula for the given geometric sequence is {f(n) = 2(-4)^{n-1}$}$.

Real-World Applications of Geometric Sequences

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.
  • Physics: Geometric sequences are used to model the motion of objects and the behavior of physical systems.

Common Mistakes to Avoid

When working with geometric sequences, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Incorrect common ratio: Make sure to identify the correct common ratio for the sequence.
  • Incorrect first term: Make sure to identify the correct first term for the sequence.
  • Incorrect formula: Make sure to use the correct formula for the sequence.
  • Incorrect exponent: Make sure to use the correct exponent for the sequence.

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 identify the common ratio of a geometric sequence?

A: To identify the common ratio, you can divide any term by its previous term. For example, if the sequence is ${2, -8, 32, -128, 512, \ldots\$}, you can divide the second term -8 by the first term 2 to get the common ratio -4.

Q: What is the formula for a geometric sequence?

A: The formula for a geometric sequence is given by {f(n) = ar^{n-1}$}$, where aa is the first term, rr is the common ratio, and nn is the term number.

Q: How do I find the formula for a geometric sequence?

A: To find the formula for a geometric sequence, you need to identify the common ratio and the first term. Once you have these values, you can plug them into the formula {f(n) = ar^{n-1}$}$ to get the formula for the sequence.

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, on the other hand, 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 determine if a sequence is geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the ratio between consecutive terms. If the ratio is constant, the sequence is geometric. If the difference between consecutive terms is constant, the sequence is arithmetic.

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

A: 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.
  • Physics: Geometric sequences are used to model the motion of objects and the behavior of physical systems.

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

A: To use geometric sequences to solve real-world problems, you need to identify the common ratio and the first term of the sequence. Once you have these values, you can plug them into the formula {f(n) = ar^{n-1}$}$ to get the formula for the sequence. You can then use this formula to solve problems related to finance, biology, computer science, and physics.

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:

  • Incorrect common ratio: Make sure to identify the correct common ratio for the sequence.
  • Incorrect first term: Make sure to identify the correct first term for the sequence.
  • Incorrect formula: Make sure to use the correct formula for the sequence.
  • Incorrect exponent: Make sure to use the correct exponent for the sequence.

Q: How do I practice working with geometric sequences?

A: To practice working with geometric sequences, you can try the following:

  • Work through examples: Try working through examples of geometric sequences to practice identifying the common ratio and the first term.
  • Solve problems: Try solving problems related to finance, biology, computer science, and physics using geometric sequences.
  • Use online resources: Use online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha to practice working with geometric sequences.

Conclusion

In conclusion, geometric sequences are an important concept in mathematics, and understanding them is crucial for many real-world applications. By identifying the common ratio and the first term, we can find the formula for a geometric sequence. With practice and experience, we can become proficient in working with geometric sequences and apply them to solve real-world problems.