Select The Correct Answer.What Is The { N $}$th Term Of The Geometric Sequence { 4, 8, 16, 32, \ldots $}$?A. { A_n = 2(4)^{n-1} $}$B. { A_n = 4(2)^{n-1} $}$C. { A_n = 2(n)^4 $} D . \[ D. \[ D . \[ A_n =
Introduction
Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will delve into the world of geometric sequences and explore the concept of the nth term of a geometric sequence. We will also examine the given options and determine the correct answer.
What is a Geometric Sequence?
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, ...
where a is the first term and r is the common ratio.
The nth Term of a Geometric Sequence
The nth term of a geometric sequence can be found using the formula:
an = ar^(n-1)
where an is the nth term, a is the first term, r is the common ratio, and n is the term number.
The Given Sequence
The given sequence is:
4, 8, 16, 32, ...
We can see that each term is obtained by multiplying the previous term by 2. Therefore, the common ratio is 2.
Option A: a_n = 2(4)^(n-1)
Let's analyze option A:
a_n = 2(4)^(n-1)
We can rewrite this as:
a_n = 2(22)(n-1)
Using the property of exponents, we can simplify this to:
a_n = 2(2^(2(n-1)))
a_n = 2^(2n-1)
This is not the correct formula for the nth term of the given sequence.
Option B: a_n = 4(2)^(n-1)
Let's analyze option B:
a_n = 4(2)^(n-1)
We can rewrite this as:
a_n = 22(2)(n-1)
Using the property of exponents, we can simplify this to:
a_n = 2^(2+n-1)
a_n = 2^(n+1)
This is the correct formula for the nth term of the given sequence.
Option C: a_n = 2(n)^4
Let's analyze option C:
a_n = 2(n)^4
This formula does not match the given sequence, and it is not a valid formula for the nth term of a geometric sequence.
Option D: a_n = 2^(n+1)
Let's analyze option D:
a_n = 2^(n+1)
This formula is similar to option B, but it is not the correct formula for the nth term of the given sequence.
Conclusion
In conclusion, the correct answer is option B: a_n = 4(2)^(n-1). This formula accurately represents the nth term of the given geometric sequence.
Key Takeaways
- 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 nth term of a geometric sequence can be found using the formula: an = ar^(n-1).
- The given sequence is a geometric sequence with a common ratio of 2.
- The correct formula for the nth term of the given sequence is: a_n = 4(2)^(n-1).
Additional Resources
For further practice and review, we recommend the following resources:
- Khan Academy: Geometric Sequences
- Mathway: Geometric Sequences
- Wolfram Alpha: Geometric Sequences
Final Thoughts
Introduction
In our previous article, we explored the concept of geometric sequences and the formula for the nth term. However, we understand that there may be more questions and concerns. In this article, we will address some of the most frequently asked questions about geometric sequences.
Q: What is the difference between a geometric sequence and an arithmetic 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. An arithmetic sequence, on the other hand, is a sequence of numbers in which each term after the first is found by adding a fixed number called the common difference.
Q: How do I find the common ratio of a geometric sequence?
A: To find the common ratio of a geometric sequence, you can divide any term by its previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can divide the second term by the first term to get 6/2 = 3. This means that the common ratio is 3.
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:
an = ar^(n-1)
where an is the nth term, a is the first term, r is the common ratio, and n is the term number.
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 for a pattern in the sequence. If the sequence is obtained by multiplying each term by a fixed number, it is a geometric sequence. If the sequence is obtained by adding a fixed number to each term, it is an arithmetic sequence.
Q: Can I have a negative common ratio in a geometric sequence?
A: Yes, you can have a negative common ratio in a geometric sequence. For example, the sequence 2, -6, 18, -54, ... has a common ratio of -3.
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:
S = a(1 - r^n) / (1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
Q: Can I have a zero common ratio in a geometric sequence?
A: No, you cannot have a zero common ratio in a geometric sequence. If the common ratio is zero, the sequence will be constant, and it will not be a geometric sequence.
Q: How do I find the nth term of a geometric sequence with a zero first term?
A: If the first term of a geometric sequence is zero, the sequence will be zero for all terms. Therefore, the nth term will also be zero.
Conclusion
In conclusion, geometric sequences are an important concept in mathematics, and understanding them is crucial for solving various problems. By mastering the concept of geometric sequences, you will be able to tackle more complex problems and develop a deeper understanding of mathematical concepts.
Additional Resources
For further practice and review, we recommend the following resources:
- Khan Academy: Geometric Sequences
- Mathway: Geometric Sequences
- Wolfram Alpha: Geometric Sequences
Final Thoughts
We hope that this article has helped to clarify any questions or concerns you may have had about geometric sequences. If you have any further questions or need additional help, please don't hesitate to ask.