Find The 7 Th 7^{\text{th}} 7 Th Term In The Sequence: − 1 , 2 , − 4 , 8 , … -1, 2, -4, 8, \ldots − 1 , 2 , − 4 , 8 , … Hint: Write A Formula To Help You.- Use The First Term And Common Ratio.- Remember To Use The Correct Order Of Operations!
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 general formula for the nth term of a geometric sequence is given by:
a_n = a_1 * 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.
Identifying the First Term and Common Ratio
In the given sequence: -1, 2, -4, 8, ..., we can identify the first term (a_1) as -1 and the common ratio (r) by dividing any term by its previous term. For example, we can divide the second term (2) by the first term (-1) to get the common ratio:
r = 2 / -1 = -2
Writing a Formula for the nth Term
Now that we have the first term (a_1 = -1) and the common ratio (r = -2), we can write a formula for the nth term of the sequence:
a_n = -1 * (-2)^(n-1)
Finding the 7th Term
To find the 7th term, we can plug in n = 7 into the formula:
a_7 = -1 * (-2)^(7-1) a_7 = -1 * (-2)^6 a_7 = -1 * 64 a_7 = -64
Therefore, the 7th term in the sequence is -64.
Order of Operations
When evaluating the formula, it's essential to follow the order of operations (PEMDAS):
- Evaluate the exponent: (-2)^6
- Multiply the result by -1
Conclusion
In this article, we learned how to find the 7th term in a geometric sequence using the formula a_n = a_1 * r^(n-1). We identified the first term (a_1 = -1) and the common ratio (r = -2) and wrote a formula for the nth term. By plugging in n = 7, we found that the 7th term in the sequence is -64.
Example Problems
- Find the 5th term in the sequence: 2, 6, 18, 54, ...
- Find the 3rd term in the sequence: -2, 4, -8, ...
- Find the 10th term in the sequence: 3, 9, 27, 81, ...
Practice Problems
- Find the 8th term in the sequence: 1, -2, 4, -8, ...
- Find the 6th term in the sequence: 2, -4, 8, -16, ...
- Find the 9th term in the sequence: -1, 2, -4, 8, ...
Solutions
- Find the 5th term in the sequence: 2, 6, 18, 54, ... a_5 = 2 * 3^(5-1) a_5 = 2 * 3^4 a_5 = 2 * 81 a_5 = 162
- Find the 3rd term in the sequence: -2, 4, -8, ... a_3 = -2 * 2^(3-1) a_3 = -2 * 2^2 a_3 = -2 * 4 a_3 = -8
- Find the 10th term in the sequence: 3, 9, 27, 81, ... a_10 = 3 * 3^(10-1) a_10 = 3 * 3^9 a_10 = 3 * 19683 a_10 = 59049
Tips and Tricks
- When working with geometric sequences, it's essential to identify the first term and the common ratio.
- Use the formula a_n = a_1 * r^(n-1) to find any term in the sequence.
- Follow the order of operations (PEMDAS) when evaluating the formula.
Real-World Applications
Geometric sequences have many real-world applications, such as:
- Compound interest: The formula for compound interest is a geometric sequence.
- Population growth: The population of a species can be modeled using a geometric sequence.
- Music: The frequency of a note in music is a geometric sequence.
Conclusion
In this article, we learned how to find the 7th term in a geometric sequence using the formula a_n = a_1 * r^(n-1). We identified the first term (a_1 = -1) and the common ratio (r = -2) and wrote a formula for the nth term. By plugging in n = 7, we found that the 7th term in the sequence is -64. We also provided example problems and solutions, as well as tips and tricks for working with geometric sequences.
Frequently Asked Questions
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: How do I find the nth term of a geometric sequence?
A: To find the nth term of a geometric sequence, you can use the formula a_n = a_1 * 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?
A: The common ratio is the fixed, non-zero number that is multiplied by each term to get the next term in the sequence.
Q: How do I find the common ratio?
A: To find the common ratio, you can divide any term by its previous term.
Q: What is the first term?
A: The first term is the first number in the sequence.
Q: How do I find the first term?
A: The first term is usually given in the problem or can be found by looking at the sequence.
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 a_n = a_1 * r^(n-1).
Q: How do I use the formula to find the nth term?
A: To use the formula, plug in the values of a_1, r, and n into the formula and simplify.
Q: What is the order of operations?
A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Q: Why is it important to follow the order of operations?
A: Following the order of operations ensures that the formula is evaluated correctly and that the correct answer is obtained.
Q: Can I use the formula to find any term in the sequence?
A: Yes, you can use the formula to find any term in the sequence by plugging in the values of a_1, r, and n into the formula.
Q: What are some real-world applications of geometric sequences?
A: Geometric sequences have many real-world applications, such as compound interest, population growth, and music.
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_n = a_1 * (1 - r^n) / (1 - r), where S_n is the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.
Q: What is the formula for the sum of a geometric sequence?
A: The formula for the sum of a geometric sequence is S_n = a_1 * (1 - r^n) / (1 - r).
Q: How do I use the formula to find the sum of a geometric sequence?
A: To use the formula, plug in the values of a_1, r, and n into the formula and simplify.
Q: What are some tips and tricks for working with geometric sequences?
A: Some tips and tricks for working with geometric sequences include:
- Identifying the first term and the common ratio
- Using the formula a_n = a_1 * r^(n-1) to find any term in the sequence
- Following the order of operations (PEMDAS)
- Using the formula S_n = a_1 * (1 - r^n) / (1 - r) to find the sum of a geometric sequence
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
- Not following the order of operations (PEMDAS)
- Not using the correct formula to find the nth term or the sum of the sequence
Q: How do I practice working with geometric sequences?
A: You can practice working with geometric sequences by:
- Working through example problems
- Creating your own geometric sequences and finding the nth term or the sum
- Using online resources or worksheets to practice
Q: What are some resources for learning more about geometric sequences?
A: Some resources for learning more about geometric sequences include:
- Online tutorials and videos
- Textbooks and workbooks
- Online communities and forums
- Math websites and apps