Find The 9th Term Of The Geometric Sequence: 5, -25, 125,
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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 a geometric sequence is:
a_n = a_1 * r^(n-1)
where:
- a_n is the nth term of the sequence
- a_1 is the first term of the sequence
- r is the common ratio
- n is the term number
Identifying the First Term and Common Ratio
In the given geometric sequence: 5, -25, 125, we can identify the first term (a_1) as 5 and the common ratio (r) by dividing the second term by the first term:
r = -25 / 5 = -5
Verifying the Common Ratio
To verify that the common ratio is indeed -5, we can divide the third term by the second term:
r = 125 / -25 = -5
Since the common ratio is consistent, we can proceed with finding the 9th term.
Finding the 9th Term
Using the formula for a geometric sequence, we can find the 9th term (a_9) by substituting the values of a_1, r, and n:
a_9 = a_1 * r^(n-1) = 5 * (-5)^(9-1) = 5 * (-5)^8 = 5 * 390625 = 1953125
Conclusion
The 9th term of the geometric sequence 5, -25, 125 is 1953125.
Example Use Case
Geometric sequences have numerous applications in real-world problems, such as:
- Finance: Compound interest is a classic example of a geometric sequence, where the interest earned on an investment is added to the principal, resulting in a new principal amount that earns interest in the next period.
- Biology: Population growth can be modeled using geometric sequences, where the population size at each time period is a fixed multiple of the previous population size.
- Computer Science: Geometric sequences are used in algorithms for image processing, where the intensity of each pixel is a fixed multiple of the intensity of the previous pixel.
Tips and Tricks
- Common Ratio: The common ratio is a crucial component of a geometric sequence. A common ratio of 1 results in an arithmetic sequence, while a common ratio of -1 results in an alternating sequence.
- Term Number: The term number (n) is used to determine the position of the term in the sequence. A term number of 1 corresponds to the first term, a term number of 2 corresponds to the second term, and so on.
- Formula: The formula for a geometric sequence is 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.
Frequently Asked Questions
- What is a geometric 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.
- How do I find the common ratio of a geometric sequence? To find the common ratio, divide the second term by the first term.
- How do I find the nth term of a geometric sequence?
To find the nth term, 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.
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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 common ratio of a geometric sequence?
A: To find the common ratio, divide the second term by the first term. For example, if the first term is 5 and the second term is -25, the common ratio would be -25 / 5 = -5.
Q: How do I find the nth term of a geometric sequence?
A: To find the nth term, 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 formula for a geometric sequence?
A: The formula for a geometric sequence is 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: How do I determine if a sequence is geometric?
A: To determine if a sequence is geometric, check if each term after the first is found by multiplying the previous term by a fixed, non-zero number. If this is the case, the sequence is geometric.
Q: What is the difference between a geometric sequence and an arithmetic sequence?
A: The main difference between a geometric sequence and an arithmetic sequence is the way each term is found. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number, while in an arithmetic sequence, each term is found by adding a fixed number to the previous term.
Q: Can a geometric sequence have a common ratio of 1?
A: Yes, a geometric sequence can have a common ratio of 1. In this case, the sequence is actually an arithmetic sequence, where each term is found by adding a fixed number to the previous term.
Q: Can a geometric sequence have a common ratio of -1?
A: Yes, a geometric sequence can have a common ratio of -1. In this case, the sequence is an alternating sequence, where each term alternates between positive and negative values.
Q: How do I find the sum of a geometric sequence?
A: To find the sum of a geometric sequence, 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 significance of the common ratio in a geometric sequence?
A: The common ratio is a crucial component of a geometric sequence, as it determines the rate at which each term increases or decreases. A common ratio of 1 results in an arithmetic sequence, while a common ratio of -1 results in an alternating sequence.
Q: Can a geometric sequence have a common ratio of 0?
A: No, a geometric sequence cannot have a common ratio of 0. This is because the common ratio is defined as the ratio of each term to the previous term, and division by zero is undefined.
Q: How do I find the product of a geometric sequence?
A: To find the product of a geometric sequence, use the formula P_n = a_1 * (1 - r^n) / (1 - r), where P_n is the product 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 difference between a geometric sequence and a harmonic sequence?
A: The main difference between a geometric sequence and a harmonic sequence is the way each term is found. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number, while in a harmonic sequence, each term is found by dividing the previous term by a fixed, non-zero number.
Q: Can a geometric sequence have a common ratio of infinity?
A: No, a geometric sequence cannot have a common ratio of infinity. This is because the common ratio is defined as the ratio of each term to the previous term, and division by zero is undefined.
Q: How do I find the reciprocal of a geometric sequence?
A: To find the reciprocal of a geometric sequence, use the formula R_n = 1 / a_n, where R_n is the reciprocal of the nth term, a_n is the nth term, and n is the term number.
Q: What is the significance of the first term in a geometric sequence?
A: The first term is a crucial component of a geometric sequence, as it determines the starting value of the sequence. The first term is used to find the common ratio and the nth term of the sequence.
Q: Can a geometric sequence have a common ratio of a fraction?
A: Yes, a geometric sequence can have a common ratio of a fraction. For example, if the common ratio is 1/2, the sequence would be 5, 2.5, 1.25, and so on.
Q: How do I find the sum of an infinite geometric sequence?
A: To find the sum of an infinite geometric sequence, use the formula S = a_1 / (1 - r), where S is the sum, a_1 is the first term, and r is the common ratio. This formula is valid only if the absolute value of the common ratio is less than 1.