Geometric Sequence Formula Finding The Nth Term
In the realm of mathematics, sequences play a fundamental role in understanding patterns and relationships between numbers. Among the various types of sequences, geometric sequences hold a special place due to their unique properties and wide range of applications. A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. This constant ratio is the defining characteristic of a geometric sequence and dictates its exponential growth or decay.
The first term of a geometric sequence, often denoted by a, serves as the foundation upon which the entire sequence is built. It's the starting point from which all subsequent terms are derived. The common ratio, denoted by r, acts as the multiplier that connects each term to its predecessor. It's the constant factor that determines the rate at which the sequence progresses.
To illustrate this concept, let's consider a simple example. Suppose we have a geometric sequence where the first term, a, is 2 and the common ratio, r, is 3. This means that the first term is 2, the second term is 2 multiplied by 3 (which is 6), the third term is 6 multiplied by 3 (which is 18), and so on. As we progress through the sequence, each term becomes three times larger than the previous one, showcasing the exponential growth characteristic of geometric sequences.
The formula for the nth term of a geometric sequence, denoted by an, provides a concise way to calculate any term in the sequence without having to manually multiply the common ratio repeatedly. This formula is expressed as:
an = a * r(n-1)
Where:
- an represents the nth term of the sequence
- a represents the first term of the sequence
- r represents the common ratio
- n represents the term number (the position of the term in the sequence)
This formula encapsulates the essence of a geometric sequence, highlighting the interplay between the first term, the common ratio, and the term number. It allows us to efficiently determine any term in the sequence, regardless of its position.
Now, let's delve into the specific sequence presented in the prompt. We are given that the first term is 4, which means a = 4. We are also given that the common ratio is -2, which means r = -2. Our goal is to determine the formula for this sequence based on the term number, n.
To achieve this, we can directly apply the formula for the nth term of a geometric sequence, substituting the given values for a and r. This yields:
an = 4 * (-2)(n-1)
This formula represents the general expression for the nth term of the sequence. It tells us that to find any term in the sequence, we simply need to raise the common ratio (-2) to the power of (n-1) and then multiply the result by the first term (4).
Let's examine how this formula works in practice. To find the first term, we substitute n = 1 into the formula:
a1 = 4 * (-2)(1-1) = 4 * (-2)0 = 4 * 1 = 4
As expected, the formula correctly gives us the first term as 4. To find the second term, we substitute n = 2 into the formula:
a2 = 4 * (-2)(2-1) = 4 * (-2)1 = 4 * (-2) = -8
This tells us that the second term of the sequence is -8. To find the third term, we substitute n = 3 into the formula:
a3 = 4 * (-2)(3-1) = 4 * (-2)2 = 4 * 4 = 16
Thus, the third term of the sequence is 16. By continuing this process, we can generate any term in the sequence using the derived formula.
The prompt presents three potential formulas for the sequence, and we need to determine which one is the correct representation. Let's analyze each formula individually:
-
an = -4 * (-2)2 - 1
This formula appears to be a constant expression, as it does not involve the term number n. This means that it would produce the same value for every term in the sequence, which contradicts the nature of a geometric sequence where terms change with the term number. Therefore, this formula is incorrect.
-
an - (4 = -2)n-1
This formula contains a syntax error, as it includes an equation (4 = -2) within the expression. This makes the formula mathematically invalid and impossible to evaluate. Therefore, this formula is also incorrect.
-
an = 4 * (-2)n
This formula closely resembles the correct formula we derived earlier, but it has a subtle difference in the exponent. Instead of (n-1), the exponent is simply n. Let's examine how this difference affects the sequence generated by the formula. If we substitute n = 1 into this formula, we get:
a1 = 4 * (-2)1 = 4 * (-2) = -8
This result contradicts the given information that the first term of the sequence is 4. Therefore, this formula is also incorrect.
Based on our analysis, the correct formula for the sequence is:
an = 4 * (-2)(n-1)
This formula accurately captures the geometric progression of the sequence, where each term is obtained by multiplying the previous term by the common ratio of -2. The exponent (n-1) ensures that the first term (when n = 1) is indeed 4, as given in the prompt.
The negative common ratio of -2 introduces an alternating pattern in the sequence. The terms alternate between positive and negative values, reflecting the sign change caused by multiplying by -2. This alternating behavior is a characteristic feature of geometric sequences with negative common ratios.
Understanding the formula for a geometric sequence is crucial for various mathematical applications, including financial calculations, population growth models, and physics problems involving exponential decay. The ability to derive and apply this formula empowers us to analyze and predict the behavior of geometric sequences in diverse scenarios.
In conclusion, the formula for the sequence with a first term of 4 and a common ratio of -2 is:
an = 4 * (-2)(n-1)
This formula accurately represents the geometric progression of the sequence and allows us to determine any term in the sequence based on its position. Understanding the derivation and application of this formula provides valuable insights into the nature of geometric sequences and their role in various mathematical and real-world contexts.
This detailed exploration has not only identified the correct formula but also provided a comprehensive understanding of the underlying concepts and principles governing geometric sequences. By delving into the derivation, analysis, and implications of the formula, we have gained a deeper appreciation for the power and elegance of mathematical sequences.
