Given That A Sequence Is Defined By The Formula $a_n = 4 \cdot (-2)^{n-1}$, Find The 20th Term.A. $a_{20} = 2,097,152$B. $a_{20} = 4,194,304$C. $a_{20} = -2,097,152$D. $a_{20} = -4,194,304$

Introduction

In mathematics, 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 formula for the nth term of 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.

In this article, we will be working with a specific geometric sequence defined by the formula:

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

Our goal is to find the 20th term of this sequence.

Understanding the Formula

Let's break down the formula and understand what it means.

• The first term, $a_1$, is given by $4 \cdot (-2)^{0} = 4$.
• The common ratio, $r$, is given by $-2$.
• The term number, $n$, is given by $20$.

Finding the 20th Term

To find the 20th term, we can plug in the values into the formula:

$$a_{20} = 4 \cdot (-2)^{20-1}$$

Simplifying the expression, we get:

$$a_{20} = 4 \cdot (-2)^{19}$$

Now, we can use the property of exponents that states $(-a)^n = a^n$ for even $n$ and $(-a)^n = -a^n$ for odd $n$. Since $19$ is an odd number, we have:

$$a_{20} = 4 \cdot (-2)^{19} = 4 \cdot -2^{19}$$

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$, we can simplify the expression further:

$$a_{20} = 4 \cdot -2^{19} = -2^{20} \cdot 2$$

Now, we can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ again:

$$a_{20} = -2^{20} \cdot 2 = -2^{21}$$

Finally, we can simplify the expression by using the fact that $2^{21} = 2 \cdot 2^{20}$:

$$a_{20} = -2^{21} = -2 \cdot 2^{20}$$

Calculating the Value

Now that we have simplified the expression, we can calculate the value of the 20th term.

$$a_{20} = -2 \cdot 2^{20} = -2 \cdot 1,048,576 = -2,097,152$$

Therefore, the 20th term of the sequence is:

$a_{20} = -2,097,152$

Conclusion

In this article, we have solved the 20th term of a geometric sequence defined by the formula $a_n = 4 \cdot (-2)^{n-1}$. We have broken down the formula, understood the properties of exponents, and simplified the expression to find the value of the 20th term. The final answer is:

$a_{20} = -2,097,152$

This result can be verified by plugging in the values into the formula and calculating the result.

References

• [1] "Geometric Sequences" by Math Open Reference. Retrieved from https://www.mathopenref.com/sequencesgeometric.html
• [2] "Exponents and Powers" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponents-and-powers/x2f-exponents-and-powers/v/exponents-and-powers

Discussion

What do you think about this problem? Do you have any questions or comments? Please feel free to share your thoughts in the discussion section below.

Related Problems

If you are interested in solving more problems like this, here are some related problems that you might find helpful:

• Find the 10th term of the sequence defined by $a_n = 3 \cdot (-4)^{n-1}$.
• Find the 15th term of the sequence defined by $a_n = 2 \cdot (-3)^{n-1}$.
• Find the 20th term of the sequence defined by $a_n = 5 \cdot (-2)^{n-1}$.

Introduction

In our previous article, we solved the 20th term of a geometric sequence defined by the formula $a_n = 4 \cdot (-2)^{n-1}$. We broke down the formula, understood the properties of exponents, and simplified the expression to find the value of the 20th term. In this article, we will answer some frequently asked questions about geometric sequences and exponents.

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 formula for the nth term of a geometric sequence?

A: The formula for the nth term of 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.

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 the previous 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 its previous term. For example, if you have the terms $a_1 = 2$ and $a_2 = 6$, you can find the common ratio by dividing $a_2$ by $a_1$:

$$r = \frac{a_2}{a_1} = \frac{6}{2} = 3$$

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 calculate the value of a term in a geometric sequence?

A: To calculate the value of a term in a geometric sequence, you can 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 property of exponents that states $(-a)^n = a^n$ for even $n$ and $(-a)^n = -a^n$ for odd $n$?

A: This property of exponents states that when you raise a negative number to an even power, the result is positive, and when you raise a negative number to an odd power, the result is negative.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following properties:

• $a^m \cdot a^n = a^{m+n}$
• $a^m \cdot b^n = (ab)^{m+n}$
• $(a^m)^n = a^{mn}$

Q: What is the difference between $2^{21}$ and $-2^{21}$?

A: $2^{21}$ is equal to $2 \cdot 2^{20}$, which is equal to $2,097,152$. $-2^{21}$, on the other hand, is equal to $-2 \cdot 2^{20}$, which is equal to $-2,097,152$.

Conclusion

In this article, we have answered some frequently asked questions about geometric sequences and exponents. We have covered topics such as the formula for the nth term of a geometric sequence, the common ratio, and the properties of exponents. We hope that this article has been helpful in clarifying any confusion you may have had about these topics.

References

• [1] "Geometric Sequences" by Math Open Reference. Retrieved from https://www.mathopenref.com/sequencesgeometric.html
• [2] "Exponents and Powers" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-exponents/x2f-exponents-and-powers/x2f-exponents-and-powers/v/exponents-and-powers

Discussion

What do you think about this article? Do you have any questions or comments? Please feel free to share your thoughts in the discussion section below.

Related Problems

If you are interested in solving more problems like this, here are some related problems that you might find helpful:

• Find the 10th term of the sequence defined by $a_n = 3 \cdot (-4)^{n-1}$.
• Find the 15th term of the sequence defined by $a_n = 2 \cdot (-3)^{n-1}$.
• Find the 20th term of the sequence defined by $a_n = 5 \cdot (-2)^{n-1}$.

I hope you find these problems helpful! If you have any questions or need further assistance, please don't hesitate to ask.