Hey guys! Today, we're diving into the fascinating world of geometric sequences. We've got a sequence here: 3, 1, 1/3, 1/9, and our mission, should we choose to accept it, is to figure out the explicit function that defines it. Buckle up, because we're about to unravel this mathematical mystery!
Understanding Geometric Sequences
Before we jump into the solution, let's make sure we're all on the same page about what a geometric sequence actually is. In layman's terms, a geometric sequence is a list of numbers where each number is found by multiplying the previous number by a constant value. This constant value is called the common ratio. Think of it like a snowball rolling down a hill – it gets bigger and bigger at a consistent rate.
To identify a geometric sequence, we need to check if there's a constant ratio between consecutive terms. Let's take our sequence: 3, 1, 1/3, 1/9. To find the common ratio, we can divide any term by its preceding term. For instance, 1 divided by 3 gives us 1/3. Similarly, (1/3) divided by 1 is also 1/3, and (1/9) divided by (1/3) is also 1/3. Bingo! We've found our common ratio: 1/3. This confirms that the sequence is indeed geometric.
Now that we've identified the common ratio, we can delve into the explicit function. An explicit function is a formula that allows us to directly calculate any term in the sequence without having to know the previous terms. It's like having a map that leads you straight to your destination, no detours needed. The general form of an explicit function for a geometric sequence is:
- f(n) = a * r^(n-1)
Where:
- f(n) represents the nth term in the sequence.
- a represents the first term in the sequence.
- r represents the common ratio.
- n represents the term number (1 for the first term, 2 for the second term, and so on).
Deciphering the Given Options
We've got two options to choose from, and only one will unlock the secret to our sequence. Let's examine each one closely:
Option A: f(n) = (1/3) * 3^(n-1)
Let's break this down. According to this function, the first term (a) would be 1/3, and the common ratio (r) would be 3. But wait a minute! Our sequence starts with 3, not 1/3. So, this option seems a bit off. While the common ratio of 3 is the reciprocal of what we expect, the first term doesn't align with our given sequence. To further solidify our understanding, let's test it out. If we plug in n = 1 (for the first term), we get f(1) = (1/3) * 3^(1-1) = (1/3) * 3^0 = (1/3) * 1 = 1/3. As we've already noticed, this doesn't match the first term in our sequence, which is 3. Therefore, we can confidently say that option A is not the correct explicit function for our sequence.
Option B: f(n) = 3 * (1/3)^(n-1)
Now, this looks more promising! Here, the first term (a) is 3, which perfectly matches our sequence. And the common ratio (r) is 1/3, which we already calculated. It seems like this option is speaking our language. To be absolutely sure, let's put it to the test. Plugging in n = 1, we get f(1) = 3 * (1/3)^(1-1) = 3 * (1/3)^0 = 3 * 1 = 3. This matches the first term in our sequence. Let's try another term, say n = 2, to see if it holds up. f(2) = 3 * (1/3)^(2-1) = 3 * (1/3)^1 = 3 * (1/3) = 1. Bingo! This also matches the second term in our sequence. We can continue testing with other values of n, and we'll find that this function accurately generates the terms of our geometric sequence.
The Verdict: Option B is the Winner!
After careful examination and testing, we can confidently conclude that the correct explicit function for the geometric sequence 3, 1, 1/3, 1/9 is:
B. f(n) = 3 * (1/3)^(n-1)
This function perfectly captures the essence of our sequence, allowing us to calculate any term with ease. It's like having the key to unlock the sequence's hidden pattern. We've successfully navigated the world of geometric sequences and emerged victorious!
Delving Deeper into Explicit Functions
Now that we've cracked the code for this specific sequence, let's take a moment to appreciate the power and versatility of explicit functions in general. An explicit function, as we've seen, provides a direct pathway to any term in a sequence. This is incredibly useful, especially when dealing with large sequences where calculating each term individually would be time-consuming and tedious. Imagine trying to find the 100th term in our sequence by repeatedly multiplying by 1/3 – it would take forever! But with the explicit function, we can simply plug in n = 100 and get the answer in a matter of seconds.
Explicit functions are also valuable tools for analyzing the behavior of sequences. By examining the formula, we can gain insights into how the sequence grows or shrinks, whether it approaches a limit, and other interesting properties. For instance, in our example, the common ratio of 1/3 tells us that the sequence is decreasing, as each term is a fraction of the previous term. As n gets larger, the terms will get closer and closer to zero. This type of analysis can be crucial in various fields, from finance to physics, where sequences are used to model real-world phenomena.
Furthermore, understanding explicit functions is a fundamental stepping stone to more advanced mathematical concepts, such as series, calculus, and differential equations. These concepts often rely on the ability to express sequences and functions in a concise and manageable form, which is exactly what explicit functions provide.
In essence, mastering explicit functions is like adding a powerful tool to your mathematical arsenal. It allows you to efficiently calculate terms, analyze sequence behavior, and build a solid foundation for future mathematical explorations.
Real-World Applications of Geometric Sequences
You might be thinking, "Okay, this is cool, but where would I ever use this in real life?" Well, geometric sequences pop up in more places than you might think! They're not just abstract mathematical concepts; they have practical applications in various fields.
Finance
One of the most common applications is in finance, specifically in calculating compound interest. When you deposit money into a savings account that earns compound interest, the interest is added to the principal, and then the next interest calculation is based on the new, larger principal. This process creates a geometric sequence where the common ratio is (1 + interest rate). Understanding geometric sequences can help you predict how your investments will grow over time.
For example, let's say you invest $1000 in an account that earns 5% interest compounded annually. The sequence of your account balance each year would be geometric: $1000, $1050, $1102.50, and so on. The explicit function for this sequence would allow you to calculate your balance after any number of years, without having to manually calculate the interest each year.
Population Growth
Geometric sequences can also be used to model population growth, particularly in scenarios where the population grows at a constant percentage rate. If a population increases by, say, 10% each year, the population size over time will form a geometric sequence. This can be a useful tool for understanding demographic trends and making predictions about future population sizes. However, it's important to note that real-world population growth is often more complex and may be influenced by other factors, such as resource availability and environmental changes.
Radioactive Decay
In the realm of physics, geometric sequences play a role in understanding radioactive decay. Radioactive substances decay at a constant rate, meaning that a certain percentage of the substance will decay over a given period of time. The amount of the substance remaining after each time period forms a geometric sequence. This principle is used in carbon dating, a technique that allows scientists to estimate the age of ancient artifacts by measuring the amount of carbon-14 remaining in the sample.
Computer Science
Geometric sequences also find applications in computer science, particularly in algorithms and data structures. For example, the binary search algorithm, which is used to efficiently search for an element in a sorted list, has a time complexity that is logarithmic, which is related to geometric sequences. The number of steps required to find an element in a list using binary search decreases geometrically with each step.
Fractals
Finally, geometric sequences are closely linked to the fascinating world of fractals. Fractals are geometric shapes that exhibit self-similarity, meaning that they look the same at different scales. Many fractals are generated using recursive processes that involve geometric sequences. For instance, the Koch snowflake, a famous fractal, is constructed by repeatedly adding equilateral triangles to the sides of a larger triangle. The number of sides and the perimeter of the snowflake increase geometrically with each iteration.
So, as you can see, geometric sequences are not just abstract mathematical concepts; they have real-world applications in various fields, from finance to physics to computer science. Understanding these sequences can give you a powerful tool for analyzing and modeling a wide range of phenomena.
Wrapping Up
So, there you have it, folks! We've successfully navigated the world of geometric sequences, deciphered the explicit function for a given sequence, and explored some real-world applications. Remember, the key to mastering these concepts is practice. So, keep exploring, keep questioning, and keep those mathematical gears turning! Who knows what other mathematical mysteries you'll unravel next?