Unraveling Arithmetic Sequences: A Step-by-Step Guide

Hey everyone! Let's dive into the fascinating world of arithmetic sequences. We'll break down different types of problems, making sure everything is super clear and easy to understand. Whether you're a math whiz or just starting out, this guide will help you master the basics and solve problems with confidence. So, let's get started, shall we?

Understanding Arithmetic Sequences

First things first: What exactly is an arithmetic sequence? An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. Think of it like climbing stairs – each step (term) is the same height (difference) apart from the last. The beauty of these sequences lies in their predictable pattern, making them relatively easy to analyze and solve. In contrast to other sequences, like geometric sequences where terms are multiplied by a common ratio, arithmetic sequences involve a simple addition or subtraction to get from one term to the next. The terms of an arithmetic sequence increase or decrease linearly, meaning they change at a constant rate. This linear behavior is a key characteristic that makes arithmetic sequences so approachable. To fully grasp arithmetic sequences, it's essential to understand the core components. You have the first term (often denoted as 'a' or 'T₁'), the common difference (d), and the nth term (Tₙ), which represents any term in the sequence. Each term in the sequence can be calculated using a simple formula: Tₙ = a + (n-1)d. This is your go-to formula for finding any term, given the first term, the common difference, and the position of the term you're looking for. It also helps to find the first term and the common difference if you know two terms in the sequence. Understanding this formula is like having a key to unlock the secrets of arithmetic sequences! Understanding this formula is like having a key to unlock the secrets of arithmetic sequences! The key to mastering arithmetic sequences is to practice solving different types of problems. Let's get our hands dirty with some problems and see how we can tackle them.

Problem Breakdown and Solutions

Okay, guys, let's get down to the nitty-gritty and work through some problems to really solidify our understanding of arithmetic sequences. Don't worry, we'll break it down step by step to ensure everyone's on the same page. We'll start with how to identify an arithmetic sequence and find its key features. Then, we'll progress to finding specific terms and solving for missing values. Each problem is designed to build your skills progressively. So, grab your pencils, and let's get started. By working through these problems together, you'll not only grasp the concepts but also build the confidence to tackle any arithmetic sequence problem that comes your way. Each step builds on the last, solidifying your understanding and enabling you to apply the principles to more complex scenarios. Ready to dive in? Let's go!

(a) Proving T_n = 3n + 5 is an Arithmetic Sequence

Alright, let's start with a classic: proving that a given sequence is arithmetic. The question asks us to show that T_n = 3n + 5 is an arithmetic sequence. How do we do this? Simple! We need to show that the difference between consecutive terms is constant. To do this, let's find the first few terms.

To find the first term (T₁), substitute n = 1 into the formula: T₁ = 3(1) + 5 = 8. For the second term (T₂), substitute n = 2: T₂ = 3(2) + 5 = 11. For the third term (T₃), substitute n = 3: T₃ = 3(3) + 5 = 14.

Now, let's find the differences between consecutive terms: T₂ - T₁ = 11 - 8 = 3 T₃ - T₂ = 14 - 11 = 3

Since the difference between consecutive terms is constant (3), this confirms that T_n = 3n + 5 is indeed an arithmetic sequence. The common difference (d) is 3. The first term (a or T₁) is 8. See? Not so hard, right?

(b) Finding the First Three Terms

Let's switch gears a bit. Now we're given some information and need to work backward to find the terms. The problem states that the 100th term of an arithmetic sequence is 98 (T₁₀₀ = 98), and the common difference is 2 (d = 2). Our mission? Find the first three terms.

We know the formula for the nth term is: Tₙ = a + (n-1)d. We can use the information about the 100th term to find the first term (a). We have T₁₀₀ = 98, n = 100, and d = 2. Plugging these values into the formula: 98 = a + (100 - 1) * 2 98 = a + 99 * 2 98 = a + 198

Now, solve for 'a': a = 98 - 198 a = -100

So, the first term (T₁) is -100. Now that we have the first term and the common difference, we can easily find the next two terms: T₂ = T₁ + d = -100 + 2 = -98 T₃ = T₂ + d = -98 + 2 = -96

Therefore, the first three terms of the sequence are -100, -98, and -96. There you have it! By knowing one term and the common difference, we can work our way back to find the beginning of the sequence. See how each problem builds on our understanding? The formula Tₙ = a + (n-1)d is your best friend here. Understanding how to manipulate this formula is a crucial skill for solving any arithmetic sequence problem.

(c) Finding Which Term Is a Specific Value

Here we go, guys! Time to solve for 'n' – i.e., determining which term in the sequence has a specific value. Let's say we have an arithmetic sequence where the first term (a) is 5 and the common difference (d) is 4. The question asks us to find which term is equal to 101 (Tₙ = 101).

We'll use the formula Tₙ = a + (n-1)d, and we know Tₙ = 101, a = 5, and d = 4. Let's plug those values in and solve for 'n': 101 = 5 + (n - 1) * 4 101 = 5 + 4n - 4 101 = 1 + 4n 100 = 4n n = 100 / 4 n = 25

So, the 25th term (T₂₅) is equal to 101. This kind of problem showcases how we can use the formula to find the position of a term if we know its value. Pretty neat, huh? Understanding how to solve for 'n' opens up another dimension of problems you can tackle in arithmetic sequences.

Conclusion: Mastering the Sequence

Alright, folks, we've covered quite a bit of ground today! We’ve taken a deep dive into arithmetic sequences, solved for missing values and confirmed specific patterns. We've shown that T_n = 3n + 5 is an arithmetic sequence and helped you feel confident to handle anything thrown your way. Remember, the key to success with arithmetic sequences is consistent practice. The more problems you solve, the more comfortable you'll become with the formulas and concepts. Keep practicing, and you'll become a pro in no time! Remember the formula Tₙ = a + (n-1)d. That's your golden ticket for solving most arithmetic sequence problems. Use it wisely, practice often, and you'll be well on your way to mastering these mathematical wonders. Keep up the excellent work, and enjoy the journey!