Simplifying the Explicit Formula

Oct 13, 2025 by windailys.com 121 views

Introduction

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 the recursive and explicit formulas for a geometric sequence, starting at 2, with a common ratio of 4.

Recursive Formula

The recursive formula for a geometric sequence is given by:

\text{NEXT} = r \times \text{NOW}

where $r$ is the common ratio and $\text{NOW}$ is the current term. In this case, the common ratio is 4, and the sequence starts at 2. Therefore, the recursive formula for this sequence is:

\text{NEXT} = 4 \times \text{NOW}

Explicit Formula

The explicit formula for a geometric sequence is given by:

a_n = a_1 \times r^{n-1}

where $a_n$ is the $n$ th term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. In this case, the first term is 2, the common ratio is 4, and we want to find the explicit formula for the sequence.

To find the explicit formula, we can start by writing out the first few terms of the sequence:

a_1 = 2

a_2 = 4 \times 2 = 8

a_3 = 4 \times 8 = 32

a_4 = 4 \times 32 = 128

We can see that each term is obtained by multiplying the previous term by 4. Therefore, the explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

Simplifying the Explicit Formula

We can simplify the explicit formula by using the property of exponents that states $a^m \times a^n = a^{m+n}$ . Therefore, we can rewrite the explicit formula as:

a_n = 2 \times 4^{n-1} = 2 \times 4^{n} \times 4^{-1} = 2 \times 4^{n} \times \frac{1}{4} = 2 \times 4^{n-1}

However, this is not the correct simplification. The correct simplification is:

a_n = 2 \times 4^{n-1} = 2 \times (4^1)^{n-1} = 2 \times 4^{n-1}

This simplification is not necessary, and the explicit formula can be left as:

a_n = 2 \times 4^{n-1}

Comparing the Options

Now that we have found the explicit formula for the sequence, we can compare it with the options given:

A. $a_n = (4)(2)^n$ B. $a_n = (8)^{n-1}$ C. $a_n = 2 \times 4^{n-1}$

We can see that option C is the correct explicit formula for the sequence.

Conclusion

In this article, we have explored the recursive and explicit formulas for a geometric sequence, starting at 2, with a common ratio of 4. We have found that the explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

We have also compared this formula with the options given and found that option C is the correct explicit formula for the sequence.

References

[1] "Geometric Sequences" by Math Open Reference
[2] "Geometric Sequences" by Purplemath

Discussion

What is a geometric sequence? How is it different from an arithmetic sequence?

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.

What is the recursive formula for a geometric sequence?

The recursive formula for a geometric sequence is given by:

\text{NEXT} = r \times \text{NOW}

where $r$ is the common ratio and $\text{NOW}$ is the current term.

What is the explicit formula for a geometric sequence?

The explicit formula for a geometric sequence is given by:

a_n = a_1 \times r^{n-1}

where $a_n$ is the $n$ th term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

What is the difference between the recursive and explicit formulas for a geometric sequence?

The recursive formula for a geometric sequence is given by:

\text{NEXT} = r \times \text{NOW}

where $r$ is the common ratio and $\text{NOW}$ is the current term.

The explicit formula for a geometric sequence is given by:

a_n = a_1 \times r^{n-1}

where $a_n$ is the $n$ th term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

The recursive formula is used to find the next term in the sequence, while the explicit formula is used to find the $n$ th term in the sequence.

What is the explicit formula for the sequence starting at 2, with a common ratio of 4?

The explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

What is the difference between the explicit formula and the options given?

The explicit formula is:

a_n = 2 \times 4^{n-1}

Option A is:

a_n = (4)(2)^n

Option B is:

a_n = (8)^{n-1}

Option C is:

a_n = 2 \times 4^{n-1}

The explicit formula is the same as option C.

What is the conclusion of this article?

In this article, we have explored the recursive and explicit formulas for a geometric sequence, starting at 2, with a common ratio of 4. We have found that the explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

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 recursive formula for a geometric sequence?

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

\text{NEXT} = r \times \text{NOW}

where $r$ is the common ratio and $\text{NOW}$ is the current term.

Q: What is the explicit formula for a geometric sequence?

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

a_n = a_1 \times r^{n-1}

where $a_n$ is the $n$ th term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: What is the difference between the recursive and explicit formulas for a geometric sequence?

A: The recursive formula is used to find the next term in the sequence, while the explicit formula is used to find the $n$ th term in the sequence.

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

A: To find the explicit formula for a geometric sequence, you need to know the first term and the common ratio. You can then use the formula:

a_n = a_1 \times r^{n-1}

to find the $n$ th term.

Q: What is the explicit formula for the sequence starting at 2, with a common ratio of 4?

A: The explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

Q: How do I compare the explicit formula with the options given?

A: To compare the explicit formula with the options given, you need to plug in the values of $a_n$ , $a_1$ , $r$ , and $n$ into the formula and see which one matches.

Q: What is the difference between the explicit formula and the options given?

A: The explicit formula is:

a_n = 2 \times 4^{n-1}

Option A is:

a_n = (4)(2)^n

Option B is:

a_n = (8)^{n-1}

Option C is:

a_n = 2 \times 4^{n-1}

The explicit formula is the same as option C.

Q: What is the conclusion of this article?

A: In this article, we have explored the recursive and explicit formulas for a geometric sequence, starting at 2, with a common ratio of 4. We have found that the explicit formula for this sequence is:

a_n = 2 \times 4^{n-1}

We have also compared this formula with the options given and found that option C is the correct explicit formula for the sequence.

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

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

Population growth: The population of a city or country can be modeled using a geometric sequence.
Finance: The value of an investment can be modeled using a geometric sequence.
Science: The growth of a population of bacteria or other organisms can be modeled using a geometric sequence.

Q: How do I use geometric sequences in real-world applications?

A: To use geometric sequences in real-world applications, you need to identify the first term and the common ratio. You can then use the formula:

a_n = a_1 \times r^{n-1}

to find the $n$ th term.

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 first term and the common ratio correctly.
Not using the correct formula to find the $n$ th term.
Not checking the units of the terms in the sequence.

Q: How do I troubleshoot common mistakes when working with geometric sequences?

A: To troubleshoot common mistakes when working with geometric sequences, you need to:

Check the units of the terms in the sequence.
Use the correct formula to find the $n$ th term.
Identify the first term and the common ratio correctly.

Q: What are some advanced topics in geometric sequences?

A: Some advanced topics in geometric sequences include:

Inverse geometric sequences.
Geometric sequences with negative common ratios.
Geometric sequences with complex common ratios.

Q: How do I learn more about geometric sequences?

A: To learn more about geometric sequences, you can:

Read books and articles on the topic.
Take online courses or watch video tutorials.
Practice solving problems and exercises.