Enter The Correct Answer In The Box.The Explicit Formula For A Certain Geometric Sequence Is F ( N ) = 525 ( 20 ) N − 1 F(n) = 525(20)^{n-1} F ( N ) = 525 ( 20 ) N − 1 . What Is The Exponential Function For The Sequence? Write Your Answer In The Form Shown.

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 explicit formula for a geometric sequence is given by:

$$f(n) = ar^{n-1}$$

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

The Explicit Formula for a Certain Geometric Sequence

The explicit formula for a certain geometric sequence is given by:

$$f(n) = 525(20)^{n-1}$$

This formula tells us that the first term of the sequence is $525$, and the common ratio is $20$.

Finding the Exponential Function for the Sequence

To find the exponential function for the sequence, we need to rewrite the explicit formula in the form:

$$f(n) = a \cdot b^n$$

where $a$ is the first term, and $b$ is the base of the exponential function.

Comparing the given explicit formula with the desired form, we can see that:

$$a = 525$$

and

$$b = 20$$

Therefore, the exponential function for the sequence is:

$$f(n) = 525 \cdot 20^n$$

Simplifying the Exponential Function

We can simplify the exponential function by combining the constants:

$$f(n) = 525 \cdot 20^n = 525 \cdot 2^{2n}$$

This is the simplified form of the exponential function for the sequence.

Conclusion

In this article, we have seen how to find the exponential function for a geometric sequence given its explicit formula. We have also seen how to simplify the exponential function by combining the constants. This is an important concept in mathematics, and it has many applications in fields such as finance, economics, and computer science.

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 explicit formula for a geometric sequence is given by $f(n) = ar^{n-1}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.
• To find the exponential function for a sequence, we need to rewrite the explicit formula in the form $f(n) = a \cdot b^n$, where $a$ is the first term, and $b$ is the base of the exponential function.
• The exponential function for a sequence can be simplified by combining the constants.

Practice Problems

1. Find the exponential function for the sequence given by $f(n) = 3 \cdot 4^n$.
2. Find the exponential function for the sequence given by $f(n) = 2 \cdot 5^n$.
3. Find the exponential function for the sequence given by $f(n) = 4 \cdot 3^n$.

Answers

1. $f(n) = 3 \cdot 4^n = 3 \cdot 2^{2n}$
2. $f(n) = 2 \cdot 5^n$
3. $f(n) = 4 \cdot 3^n = 4 \cdot 3^n$

References

• [1] "Geometric Sequences and Series" by Math Open Reference
• [2] "Exponential Functions" by Khan Academy
• [3] "Geometric Sequences and Exponential Functions" by Wolfram MathWorld
  Geometric Sequences and Exponential Functions: Q&A =====================================================

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: What is the explicit formula for a geometric sequence?

A: The explicit formula for a geometric sequence is given by:

$$f(n) = ar^{n-1}$$

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

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

A: To find the exponential function for a sequence, we need to rewrite the explicit formula in the form:

$$f(n) = a \cdot b^n$$

where $a$ is the first term, and $b$ is the base of the exponential function.

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

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 exponential function is a mathematical function of the form $f(x) = a \cdot b^x$, where $a$ is the first term, and $b$ is the base of the exponential function.

Q: Can I simplify the exponential function for a geometric sequence?

A: Yes, you can simplify the exponential function for a geometric sequence by combining the constants. For example, if the explicit formula for a geometric sequence is $f(n) = 525 \cdot 20^n$, you can simplify it to $f(n) = 525 \cdot 2^{2n}$.

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

A: To find the first term of a geometric sequence, you need to know the explicit formula for the sequence. The first term is the value of $a$ in the explicit formula.

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

A: To find the common ratio of a geometric sequence, you need to know the explicit formula for the sequence. The common ratio is the value of $r$ in the explicit formula.

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

A: Yes, geometric sequences can be used to model real-world phenomena such as population growth, financial investments, and chemical reactions.

Q: What are some common applications of geometric sequences and exponential functions?

A: Some common applications of geometric sequences and exponential functions include:

• Modeling population growth and decline
• Calculating compound interest and investment returns
• Analyzing chemical reactions and radioactive decay
• Modeling the spread of diseases and epidemics
• Analyzing the growth of companies and economies

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

A: To use geometric sequences and exponential functions in real-world problems, you need to:

• Identify the problem and the variables involved
• Determine the type of sequence or function needed to model the problem
• Use the explicit formula or exponential function to calculate the desired value
• Interpret the results and make conclusions based on the data

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

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

• Confusing the explicit formula with the exponential function
• Failing to simplify the exponential function
• Using the wrong values for the first term and common ratio
• Failing to interpret the results correctly

Q: How do I practice working with geometric sequences and exponential functions?

A: To practice working with geometric sequences and exponential functions, you can:

• Work through practice problems and exercises
• Use online resources and calculators to explore and visualize the concepts
• Apply the concepts to real-world problems and scenarios
• Join online communities and forums to discuss and learn from others.