What Are The First Three Terms Of A Geometric Sequence Where The Fifth Term Is 25 And The Common Ratio Is 5?
In the realm of mathematics, geometric sequences hold a special place due to their elegant structure and predictable behavior. These sequences are characterized by a constant ratio between consecutive terms, a property that allows us to extrapolate and interpolate values with ease. In this article, we embark on a journey to decipher the initial terms of a geometric sequence, given the fifth term and the common ratio. Understanding geometric sequences is not just a mathematical exercise; it's a fundamental concept that finds applications in various fields, from finance and economics to physics and computer science. The ability to determine the initial terms of a geometric sequence opens doors to a deeper understanding of exponential growth and decay, essential concepts in modeling real-world phenomena.
To truly grasp the essence of geometric sequences, let's first delve into their core definition. A geometric sequence is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This constant ratio, often denoted by 'r', dictates the rate at which the sequence grows or shrinks. The general form of a geometric sequence can be expressed as: a, ar, ar^2, ar^3, ..., where 'a' represents the first term and 'r' is the common ratio. The nth term of the sequence, denoted as a_n, can be calculated using the formula: a_n = a * r^(n-1). This formula serves as a cornerstone for analyzing and manipulating geometric sequences.
Now, let's consider the specific problem at hand: determining the first three terms of a geometric sequence where the fifth term (a_5) is 25 and the common ratio (r) is 5. Our mission is to unravel the mystery of the initial terms that give rise to this sequence. To achieve this, we'll employ the formula for the nth term of a geometric sequence, strategically substituting the known values to solve for the unknown first term. Once we've identified the first term, we can readily calculate the subsequent terms by successively multiplying by the common ratio. This step-by-step approach will illuminate the path to unveiling the initial terms of the sequence. As we navigate through the solution, we'll not only arrive at the answer but also gain a deeper appreciation for the interplay between terms in a geometric sequence.
To find the first three terms of the geometric sequence, we'll embark on a step-by-step journey, leveraging the fundamental properties of geometric sequences and the given information. Our primary tool will be the formula for the nth term of a geometric sequence: a_n = a * r^(n-1), where a_n represents the nth term, a is the first term, r is the common ratio, and n is the term number. In our case, we know that a_5 = 25 and r = 5. Our initial goal is to determine the value of 'a', the elusive first term.
Step 1: Unveiling the First Term
We begin by substituting the known values into the formula for the nth term. Since we know a_5 = 25 and r = 5, we can write the equation as: 25 = a * 5^(5-1). Simplifying the exponent, we get: 25 = a * 5^4. Further calculation reveals that 5^4 equals 625, transforming our equation into: 25 = a * 625. To isolate 'a', we divide both sides of the equation by 625, resulting in: a = 25 / 625. Simplifying this fraction, we arrive at the first term: a = 1/25. With the first term now unveiled, we've taken the crucial first step in decoding the geometric sequence. This value serves as the foundation upon which we'll build the subsequent terms.
Step 2: Tracing the Second Term
With the first term in hand, we can now set our sights on determining the second term of the sequence. Recall that in a geometric sequence, each term is obtained by multiplying the preceding term by the common ratio. Therefore, to find the second term, we multiply the first term (1/25) by the common ratio (5): (1/25) * 5. This calculation yields: 5/25, which simplifies to 1/5. Thus, the second term of the geometric sequence is 1/5. The consistent application of the common ratio allows us to seamlessly transition from one term to the next.
Step 3: Revealing the Third Term
Following the same principle, we now turn our attention to the third term. We obtain the third term by multiplying the second term (1/5) by the common ratio (5): (1/5) * 5. This straightforward calculation results in 1. Consequently, the third term of the geometric sequence is 1. With the third term now revealed, we've successfully traced the first three terms of the sequence, piecing together the initial segment of this geometric progression. The consistent application of the common ratio has allowed us to unveil the hidden pattern within the sequence.
Having meticulously calculated the first three terms, we now present the solution in its entirety. The first three terms of the geometric sequence are: 1/25, 1/5, and 1. This sequence embodies the essence of a geometric progression, where each term is precisely 5 times the previous term. The initial term, 1/25, sets the stage for the subsequent terms to unfold, guided by the constant influence of the common ratio. The second term, 1/5, represents the first step in the sequence's growth, while the third term, 1, marks a significant milestone in the progression.
Comparing our solution to the given options, we find that option A: 1/25, 1/5, 1, perfectly matches our calculated terms. Therefore, option A is the correct answer. The other options can be readily dismissed as they do not conform to the pattern dictated by the common ratio of 5. Option B, 25, 125, 625, represents a geometric sequence with a common ratio of 5, but it does not align with the condition that the fifth term is 25. Option C, 1/25, 1/125, 1/625, represents a geometric sequence with a common ratio of 1/5, which contradicts the given common ratio of 5. Thus, by systematically applying the principles of geometric sequences and comparing our solution to the provided options, we confidently affirm that option A is the definitive answer.
Geometric sequences, like the one we've just explored, are more than just a mathematical curiosity; they are a fundamental concept that permeates various branches of mathematics and finds applications in numerous real-world scenarios. The beauty of geometric sequences lies in their inherent predictability, governed by the constant common ratio that dictates the progression of terms. This predictability allows us to model phenomena that exhibit exponential growth or decay, such as compound interest, population growth, and radioactive decay. Understanding geometric sequences equips us with the tools to analyze and predict the behavior of these systems, making it an invaluable asset in fields like finance, economics, and physics.
The formula for the nth term of a geometric sequence, a_n = a * r^(n-1), is a powerful tool that allows us to calculate any term in the sequence, given the first term and the common ratio. This formula not only simplifies the process of finding specific terms but also provides insights into the overall structure of the sequence. For instance, we can readily observe that if the common ratio is greater than 1, the sequence will exhibit exponential growth, with terms increasing rapidly as n increases. Conversely, if the common ratio is between 0 and 1, the sequence will exhibit exponential decay, with terms approaching zero as n increases. These observations highlight the crucial role of the common ratio in shaping the behavior of a geometric sequence.
Beyond individual terms, geometric sequences also lend themselves to the calculation of sums of terms. The sum of the first n terms of a geometric sequence, denoted as S_n, can be calculated using the formula: S_n = a * (1 - r^n) / (1 - r), provided that r is not equal to 1. This formula is particularly useful in scenarios where we need to determine the cumulative effect of a geometric progression, such as the total amount accumulated in a savings account with compound interest or the total distance traveled by a bouncing ball. The ability to calculate sums of geometric sequences further enhances their applicability in solving real-world problems.
The elegance and predictability of geometric sequences extend far beyond the confines of the mathematics classroom. They are powerful tools for modeling and understanding a wide range of phenomena in the real world. One of the most common applications of geometric sequences lies in the realm of finance, where they are used to calculate compound interest. Compound interest is the interest earned not only on the principal amount but also on the accumulated interest from previous periods. This creates a geometric progression, where the balance in the account grows exponentially over time. The formula for compound interest is directly derived from the principles of geometric sequences, allowing us to accurately predict the growth of investments and loans.
Another significant application of geometric sequences is in the field of population growth. In many populations, the growth rate is proportional to the current population size. This leads to exponential growth, which can be modeled using a geometric sequence. By understanding the common ratio, which represents the growth rate, we can project future population sizes and make informed decisions about resource allocation and planning. Geometric sequences also play a crucial role in modeling radioactive decay. Radioactive isotopes decay at a constant rate, meaning that the amount of the isotope decreases exponentially over time. This decay process can be modeled using a geometric sequence, allowing us to determine the half-life of a radioactive substance and predict its decay rate.
Furthermore, geometric sequences find applications in various other fields, including computer science, physics, and engineering. In computer science, they are used in the analysis of algorithms and data structures, particularly in scenarios involving exponential time or space complexity. In physics, they are used to model damped oscillations and other phenomena where the amplitude decreases exponentially over time. In engineering, they are used in the design of filters and other systems that exhibit exponential behavior. The versatility of geometric sequences makes them an indispensable tool for scientists, engineers, and mathematicians alike.
In conclusion, geometric sequences are a fundamental concept in mathematics with far-reaching applications in the real world. Their predictable nature, governed by the constant common ratio, makes them invaluable tools for modeling exponential growth and decay phenomena. By understanding the principles of geometric sequences, we can unravel the mysteries of compound interest, population growth, radioactive decay, and a myriad of other real-world processes. The ability to determine the terms of a geometric sequence, calculate their sums, and apply these concepts to practical problems is a testament to the power and versatility of this mathematical tool. As we've seen in this article, the journey of deciphering the initial terms of a geometric sequence is not just an exercise in algebra; it's a gateway to a deeper understanding of the mathematical world and its connection to the world around us.