Finding Explicit Formula Of Geometric Sequence With Fourth Term -1 And Common Ratio -1

Hey guys! Let's dive into a cool math problem today that involves geometric sequences. We're going to figure out how to find the explicit formula for a sequence when we know a specific term and the common ratio. It might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand. So, grab your thinking caps, and let’s get started!

Understanding Geometric Sequences

Before we jump into the problem, let's make sure we're all on the same page about what a geometric sequence actually is. A geometric sequence is simply a list of numbers where each term is found by multiplying the previous term 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 really nail this down, consider this. In a geometric sequence, you always multiply by the same number to get to the next term. For example, if we start with 2 and multiply by 3 each time, we get the sequence 2, 6, 18, 54, and so on. Here, 3 is our common ratio. Understanding this basic principle is crucial because it's the foundation for finding the explicit formula. The explicit formula will allow us to find any term in the sequence without having to list out all the terms before it. It’s like having a magic key that unlocks any term we want!

Let's break down the key terms further. The first term is usually denoted as a₁, and the common ratio as 'r'. So, if you have a sequence like 4, 8, 16, 32..., a₁ is 4, and r (the number we multiply by each time) is 2. Recognizing these components is the first step in tackling any geometric sequence problem. Remember, geometric sequences are all about this consistent multiplication, so keep that idea in mind as we move forward.

The Explicit Formula: Our Secret Weapon

Now, let's talk about the star of the show – the explicit formula. This formula is like a secret weapon that allows us to calculate any term in a geometric sequence directly. It's written as:

aₙ = a₁ * r^(n-1)

Where:

• aₙ is the nth term (the term we want to find)
• a₁ is the first term of the sequence
• r is the common ratio
• n is the term number (e.g., 1 for the first term, 2 for the second term, etc.)

This formula might look a little intimidating at first, but it’s actually quite straightforward. The explicit formula essentially says that any term in the sequence is equal to the first term multiplied by the common ratio raised to the power of (n-1). Let's break down why it works this way. Starting from the first term (a₁), to get to the second term, you multiply by r once. To get to the third term, you multiply by r twice (which is r²), and so on. So, to get to the nth term, you multiply by r a total of n-1 times, hence the r^(n-1) in the formula.

The power of the explicit formula is that it saves us a ton of time. Imagine you want to find the 100th term of a sequence. Listing out all 99 terms before it would take ages! But with the explicit formula, you just plug in the values for a₁, r, and n (in this case, 100), and bam! You have your answer. This formula is a game-changer, and mastering it is key to handling geometric sequence problems like a pro. So, keep this formula handy, because we’re going to use it to solve our problem.

Problem Breakdown: Unpacking the Details

Alright, let's get to the problem we're tackling today. The question states: “If the fourth term in a geometric sequence is -1 and the common ratio is -1, find the explicit formula of the sequence.” Let's break this down piece by piece to make sure we fully grasp what we're dealing with.

The first key piece of information is that the fourth term is -1. In mathematical notation, this is written as a₄ = -1. This tells us the value of a specific term in the sequence, which is a crucial piece of the puzzle. We also know that the common ratio is -1. This means that r = -1. Remember, the common ratio is the number we multiply by to get from one term to the next. In this case, multiplying by -1 will alternate the sign of the terms.

So, what are we trying to find? The question asks us to find the explicit formula of the sequence. This means we need to find a formula that will give us any term (aₙ) in the sequence, based on its position (n). In other words, we need to determine the values of a₁ and r so that we can plug them into the general explicit formula aₙ = a₁ * r^(n-1). We already know r = -1, so the main challenge is to find a₁, the first term. Once we have a₁, we can write out the complete explicit formula and solve the problem. The explicit formula is our ultimate goal here, as it will define the entire sequence.

Finding the First Term (a₁)

Now comes the fun part – finding the first term (a₁). We know that a₄ = -1 and r = -1. We can use the explicit formula to work backward and find a₁. Here's how:

We start with the explicit formula: aₙ = a₁ * r^(n-1). Since we know a₄ = -1, we can plug in n = 4 into the formula: a₄ = a₁ * r^(4-1). Now we can substitute the values we know: -1 = a₁ * (-1)^(4-1). This simplifies to -1 = a₁ * (-1)³.

Let's break down the math. (-1)³ means -1 multiplied by itself three times, which is -1 * -1 * -1 = -1. So our equation becomes -1 = a₁ * -1. To solve for a₁, we need to isolate it. We can do this by dividing both sides of the equation by -1: -1 / -1 = (a₁ * -1) / -1. This simplifies to 1 = a₁. So, we've found it! The first term of the sequence (a₁) is 1.

Finding a₁ is a critical step in determining the explicit formula. By using the information we had about a₄ and r, and applying the explicit formula in reverse, we were able to unlock the value of a₁. This shows how versatile the explicit formula is – it's not just for finding terms, but also for finding missing pieces of the sequence. Now that we know both a₁ and r, we are ready to construct the complete explicit formula.

Constructing the Explicit Formula

With a₁ = 1 and r = -1 in our toolkit, we're ready to build the explicit formula for this geometric sequence. Remember, the general form of the explicit formula is:

aₙ = a₁ * r^(n-1)

Now, we simply substitute the values we've found. Replacing a₁ with 1 and r with -1, we get:

aₙ = 1 * (-1)^(n-1)

This is the explicit formula for our geometric sequence! It tells us how to find any term in the sequence. For instance, if we wanted to find the 5th term (a₅), we would plug in n = 5: a₅ = 1 * (-1)^(5-1) = 1 * (-1)⁴ = 1 * 1 = 1. So the 5th term is 1.

To simplify our formula further, we can recognize that multiplying by 1 doesn't change anything, so we can write the formula as:

aₙ = (-1)^(n-1)

This simplified form is even cleaner and easier to use. This explicit formula is the key to understanding this sequence. It encapsulates the entire pattern of the sequence in a single, elegant equation. Now, let’s compare our result with the options given in the original problem to select the correct answer.

Matching the Formula to the Options

Okay, we've derived the explicit formula for the geometric sequence: aₙ = 1 * (-1)^(n-1) or simply aₙ = (-1)^(n-1). Now, let's match this with the options provided in the original problem. The options were:

A) aₙ = (-1) * (-1)^(n-1) for n < 1 B) aₙ = (-1) * (1)^(n-1) for n ≥ 1 C) aₙ = (1) * (1)^(n-1) for n ≥ 2

Let's analyze each option to see which one matches our derived formula:

• Option A: aₙ = (-1) * (-1)^(n-1) for n < 1. This formula has an extra factor of -1 multiplied at the beginning, which doesn't match our formula. Also, the condition n < 1 is unusual for sequences, as n usually represents the term number, which starts from 1. So, this option is not correct.
• Option B: aₙ = (-1) * (1)^(n-1) for n ≥ 1. Here, the common ratio is 1, not -1, so this doesn't fit the problem conditions. Anything raised to the power of 1 will always be 1, so this would result in a constant sequence, not a geometric sequence with a common ratio of -1. So, this option is incorrect.
• Option C: aₙ = (1) * (1)^(n-1) for n ≥ 2. Similar to option B, the common ratio here is 1, which is incorrect. Also, the condition n ≥ 2 is not ideal, as we want a formula that works for all n starting from 1. So, this option is also incorrect.

It seems like none of the provided options perfectly match our derived formula aₙ = 1 * (-1)^(n-1). However, if we look closely, our formula is equivalent to aₙ = (-1)^(n-1). This form is not explicitly listed, but it's the correct representation of the sequence. Sometimes, multiple forms can represent the same sequence, and it’s crucial to understand the equivalencies. In this case, we can see that none of the options perfectly reflect our answer, highlighting the importance of understanding the underlying math rather than just pattern matching.

Conclusion: Mastering Geometric Sequences

So, we've successfully navigated through this geometric sequence problem! We started with knowing the fourth term and the common ratio, and we worked our way to finding the explicit formula for the sequence. Remember, the explicit formula is our powerful tool for describing any geometric sequence, allowing us to find any term without listing out the entire sequence. The explicit formula aₙ = a₁ * r^(n-1) is a fundamental concept in sequences, and understanding how to use it is key to solving these types of problems.

We broke down the problem step-by-step, emphasizing the importance of understanding the definitions and applying the formulas correctly. We also saw how to work backward using the formula to find missing pieces, like the first term (a₁). Furthermore, we compared our solution with the given options and discussed why some options were incorrect, reinforcing the critical thinking needed in math problem-solving. Even though none of the provided options perfectly matched our derived formula, we understood the underlying math to recognize the correct representation of the sequence.

Geometric sequences might seem intimidating at first, but with practice and a solid understanding of the formulas and concepts, you can tackle them with confidence. Keep practicing, and you'll become a geometric sequence master in no time! Remember, math is all about understanding the process and applying the right tools. Keep exploring, keep learning, and most importantly, have fun with it!