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: F ( N ) = A R ( R ) N F(n)=\frac{a}{r}(r)^n F ( N ) = R A ​ ( R ) N

Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding their explicit formula and exponential function is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will delve into the explicit formula for a certain geometric sequence, $f(n)=525(20)^{n-1}$, and derive the exponential function for the sequence in the form $f(n)=\frac{a}{r}(r)^n$.

Understanding Geometric Sequences

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The general form of a geometric sequence is $a, ar, ar^2, ar^3, \ldots$, where $a$ is the first term and $r$ is the common ratio.

Explicit Formula for Geometric Sequences

The explicit formula for a geometric sequence is given by $f(n)=a(r)^{n-1}$, where $a$ is the first term and $r$ is the common ratio. This formula allows us to find the $n$th term of the sequence without having to find each term individually.

Deriving the Exponential Function

To derive the exponential function for the sequence, we need to rewrite the explicit formula in the form $f(n)=\frac{a}{r}(r)^n$. We can do this by multiplying both sides of the explicit formula by $\frac{1}{r}$.

$$f(n)=a(r)^{n-1}$$

$$f(n)=a(r)^{n-1}\cdot\frac{1}{r}$$

$$f(n)=\frac{a}{r}(r)^n$$

Applying the Derivation to the Given Sequence

Now that we have derived the exponential function for the sequence, we can apply it to the given sequence, $f(n)=525(20)^{n-1}$. We can rewrite the explicit formula in the form $f(n)=\frac{a}{r}(r)^n$ by multiplying both sides by $\frac{1}{20}$.

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

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

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

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

$$f(n)=26.25(20)^n$$

Conclusion

In this article, we have derived the exponential function for a geometric sequence in the form $f(n)=\frac{a}{r}(r)^n$. We have applied this derivation to the given sequence, $f(n)=525(20)^{n-1}$, and obtained the exponential function $f(n)=26.25(20)^n$. This demonstrates the importance of understanding the explicit formula and exponential function for geometric sequences in solving various problems in mathematics.

Key Takeaways

• Geometric sequences are a fundamental concept in mathematics.
• The explicit formula for a geometric sequence is $f(n)=a(r)^{n-1}$.
• The exponential function a geometric sequence is $f(n)=\frac{a}{r}(r)^n$.
• To derive the exponential function, multiply both sides of the explicit formula by $\frac{1}{r}$.
• The given sequence, $f(n)=525(20)^{n-1}$, can be rewritten in the form $f(n)=\frac{a}{r}(r)^n$ by multiplying both sides by $\frac{1}{20}$.

Further Reading

For more information on geometric sequences and their applications, we recommend the following resources:

• Khan Academy: Geometric Sequences
• Math Is Fun: Geometric Sequences
• Wolfram MathWorld: Geometric Sequence

References

• [1] Larson, R. (2019). Elementary Algebra. Cengage Learning.
• [2] Sullivan, M. (2018). College Algebra. Pearson Education.
• [3] Anton, H. (2017). Calculus: Early Transcendentals. John Wiley & Sons.
  Geometric Sequences: Q&A ==========================

Q: What is a geometric sequence?

A: A geometric sequence is a sequence of numbers in which 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 $f(n)=a(r)^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

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

A: To find the exponential function for a geometric sequence, multiply both sides of the explicit formula by $\frac{1}{r}$. This will give you the exponential function in the form $f(n)=\frac{a}{r}(r)^n$.

Q: Can you give an example of how to find the exponential function for a geometric sequence?

A: Let's say we have the geometric sequence $f(n)=525(20)^{n-1}$. To find the exponential function, we can multiply both sides by $\frac{1}{20}$.

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

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

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

$$f(n)=26.25(20)^n$$

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the fixed, non-zero number that is multiplied by each term to get the next term.

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

A: To find the common ratio in a geometric sequence, you can divide any term by the previous term. For example, if we have the geometric sequence $2, 6, 18, 54, \ldots$, we can find the common ratio by dividing the second term by the first term.

$$\frac{6}{2}=3$$

So, the common ratio is $3$.

Q: Can you give an example of how to find the common ratio in a geometric sequence?

A: Let's say we have the geometric sequence $3, 9, 27, 81, \ldots$. To find the common ratio, we can divide the second term by the first term.

$$\frac{9}{3}=3$$

So, the common ratio is $3$.

Q: What is the first term in a geometric sequence?

A: The first term in a geometric sequence is the first number in the sequence.

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

A: To find the first term in a geometric sequence, you can look at the first number in the sequence. For example, if we have the geometric sequence $2, 6, 18, 54, \ldots$, the first term is $2$.

Q: Can you give an example of how to find the first term in a geometric sequence?

A: Let's say we have the geometric sequence $, 12, 36, 108, \ldots$. To find the first term, we can look at the first number in the sequence.

The first term is $4$.

Q: What is the nth term in a geometric sequence?

A: The nth term in a geometric sequence is the nth number in the sequence.

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

A: To find the nth term in a geometric sequence, you can use the explicit formula $f(n)=a(r)^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Q: Can you give an example of how to find the nth term in a geometric sequence?

A: Let's say we have the geometric sequence $2, 6, 18, 54, \ldots$ and we want to find the 5th term. We can use the explicit formula to find the 5th term.

$$f(5)=2(3)^{5-1}$$

$$f(5)=2(3)^4$$

$$f(5)=2(81)$$

$$f(5)=162$$

So, the 5th term is $162$.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence is the sum of all the terms in the sequence.

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

A: To find the sum of a geometric sequence, you can use the formula for the sum of a geometric series, which is $S_n=\frac{a(1-r^n)}{1-r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: Can you give an example of how to find the sum of a geometric sequence?

A: Let's say we have the geometric sequence $2, 6, 18, 54, \ldots$ and we want to find the sum of the first 5 terms. We can use the formula for the sum of a geometric series to find the sum.

$$S_5=\frac{2(1-3^5)}{1-3}$$

$$S_5=\frac{2(1-243)}{-2}$$

$$S_5=\frac{2(-242)}{-2}$$

$$S_5=242$$

So, the sum of the first 5 terms is $242$.

Conclusion

In this article, we have answered some common questions about geometric sequences, including how to find the explicit formula, exponential function, common ratio, first term, nth term, and sum of a geometric sequence. We hope this article has been helpful in understanding geometric sequences and how to work with them.