Finding The Sum Of A 9 Term Geometric Sequence First Term 4 Last Term 1024

In the realm of mathematics, geometric sequences hold a significant position, particularly in areas like calculus, financial mathematics, and computer science. A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This contrasts with arithmetic sequences, where terms are generated by adding a constant difference. Understanding geometric sequences is crucial for solving various mathematical problems, including those involving sums of terms within a sequence.

This article delves into the method of finding the sum of a 9-term geometric sequence, given the first term and the last term. This is a common problem type in algebra and precalculus courses, testing a student's understanding of geometric sequence formulas and their applications. We will explore the formula for the sum of a finite geometric series and apply it to the specific case where the first term is 4, the last term is 1,024, and the number of terms is 9. By understanding the underlying principles and the steps involved, readers will be equipped to solve similar problems efficiently and accurately.

A geometric sequence is characterized by a constant ratio between successive terms. This constant ratio, denoted as 'r', is the cornerstone of the sequence. The first term is typically denoted as 'a', and each subsequent term is obtained by multiplying the previous term by 'r'. Thus, the sequence takes the form: a, ar, ar^2, ar^3, and so on. The nth term of a geometric sequence can be expressed as a_n = a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number. For instance, in the sequence 2, 6, 18, 54, the first term (a) is 2, and the common ratio (r) is 3. Each term is obtained by multiplying the previous term by 3. Identifying the first term and the common ratio is essential for working with geometric sequences.

The properties of geometric sequences make them particularly useful in various mathematical applications. One key property is the formula for the sum of the first n terms of a geometric sequence, which is given by: S_n = a * (1 - r^n) / (1 - r), provided r ≠ 1. This formula allows us to efficiently calculate the sum of a finite number of terms in a geometric sequence without having to add each term individually. When r = 1, the sequence becomes a constant sequence, and the sum is simply n * a. Geometric sequences also exhibit exponential growth or decay, depending on whether the absolute value of r is greater than or less than 1, respectively. Understanding these properties is crucial for solving problems related to geometric sequences and their sums.

Consider a geometric sequence with 9 terms. The first term, denoted as 'a', is given as 4, and the last term, denoted as a_9, is given as 1,024. The objective is to find the sum of all 9 terms in this geometric sequence. This problem combines the fundamental concepts of geometric sequences with the application of the sum formula. To solve this problem, we need to determine the common ratio 'r' and then use the formula for the sum of a finite geometric series. This involves understanding how the terms of a geometric sequence are related and how to manipulate the relevant formulas to find the desired sum. The problem highlights the importance of recognizing the pattern in geometric sequences and using the appropriate formulas to solve related problems. The challenge lies in efficiently finding the common ratio and applying the sum formula correctly to obtain the accurate result.

To find the sum of the 9-term geometric sequence, the first step is to determine the common ratio, 'r'. We know the first term (a) is 4 and the last term (a_9) is 1,024. The formula for the nth term of a geometric sequence is a_n = a * r^(n-1). In this case, we have a_9 = a * r^(9-1), which simplifies to 1,024 = 4 * r^8. To solve for 'r', we first divide both sides of the equation by 4, which gives us r^8 = 256. Taking the eighth root of both sides, we get r = ±2. This indicates that there are two possible values for the common ratio: 2 and -2. This is because raising either 2 or -2 to the power of 8 will result in 256. Understanding that both positive and negative values are possible is crucial for ensuring we consider all possible scenarios. The existence of two possible values for the common ratio means there might be two different geometric sequences that satisfy the given conditions, each leading to a potentially different sum.

Now that we have found the possible values for the common ratio (r = 2 and r = -2), we can calculate the sum of the 9-term geometric sequence for each case. The formula for the sum of the first n terms of a geometric sequence is S_n = a * (1 - r^n) / (1 - r). For r = 2, we have: S_9 = 4 * (1 - 2^9) / (1 - 2). Calculating 2^9 gives us 512, so the equation becomes S_9 = 4 * (1 - 512) / (-1). Simplifying further, we get S_9 = 4 * (-511) / (-1), which equals 2,044. This means that when the common ratio is 2, the sum of the 9 terms is 2,044. For r = -2, we use the same formula: S_9 = 4 * (1 - (-2)^9) / (1 - (-2)). Calculating (-2)^9 gives us -512, so the equation becomes S_9 = 4 * (1 - (-512)) / (1 + 2). Simplifying further, we get S_9 = 4 * (1 + 512) / 3, which equals 4 * 513 / 3, resulting in S_9 = 684. However, since 684 is not among the provided answer choices, we focus on the result obtained with r = 2.

Based on the calculations performed, we found that when the common ratio is 2, the sum of the 9-term geometric sequence is 2,044. This result aligns with one of the answer choices provided in the problem. The sum was calculated using the formula for the sum of a finite geometric series, S_n = a * (1 - r^n) / (1 - r), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. The given information included the first term (4), the last term (1,024), and the number of terms (9). By determining the common ratio and applying the formula, we arrived at the sum of 2,044. This process demonstrates the importance of understanding the properties of geometric sequences and the correct application of relevant formulas. The correct answer, therefore, is B. 2,044.

In conclusion, finding the sum of a geometric sequence involves understanding the sequence's properties and applying the appropriate formulas. In this case, we successfully calculated the sum of a 9-term geometric sequence with the first term being 4 and the last term being 1,024. The key steps included determining the common ratio, which was found to be 2, and then applying the formula for the sum of a finite geometric series. The calculated sum, 2,044, matched one of the provided answer choices, confirming the correctness of our approach. This problem underscores the importance of recognizing geometric sequences, understanding their formulas, and applying them accurately. Geometric sequences are a fundamental concept in mathematics, with applications in various fields, making it essential to master the techniques for solving related problems. This exercise not only enhances problem-solving skills but also reinforces the understanding of mathematical concepts and their practical applications.