Arithmetic Sequence: Find Terms And Differences

Hey guys! Let's dive into the fascinating world of arithmetic sequences. We'll tackle a problem where we need to find the difference, the first term, the 40th term, and a specific term number, given some clues about the sequence. It's like a math detective game, and we're the detectives! This guide is designed to be super friendly and easy to follow, so even if you're new to this, you'll get the hang of it.

Understanding Arithmetic Sequences

First things first, what exactly is an arithmetic sequence? Well, it's a list of numbers where the difference between any two consecutive terms is constant. Think of it like climbing stairs – each step (the difference) is the same height. This constant difference is super important; we call it the 'common difference,' often denoted by the letter 'd'. Each term in the sequence can be calculated by adding or subtracting 'd' from the previous term. The general form of an arithmetic sequence is: a, a + d, a + 2d, a + 3d, and so on. Where 'a' is the first term.

So, if we know the first term ('a') and the common difference ('d'), we can find any term in the sequence. For example, if a = 2 and d = 3, our sequence would be: 2, 5, 8, 11... See how we're just adding 3 each time?

Now, let's look at the given problem. We know that the 4th term (let's call it a₄) is 14, and the 9th term (a₉) is 9. Our mission is to find the common difference (d), the first term (a₁ or simply 'a'), the 40th term (a₄₀), and, just for fun, the term number if the value is a specific number. Sounds like a lot, but don't worry, we'll break it down step by step, and it'll be a piece of cake. The key to solving problems like these is to understand the relationship between the terms and the common difference. We're essentially working with linear equations here. Also, there are many real-world applications of arithmetic sequences, like calculating the installments of a loan, predicting the growth of something at a constant rate, and more, so what we're learning here isn't just about solving a math problem, it has practical uses too!

Finding the Common Difference (d)

Okay, let's start with the common difference, 'd'. We know two terms: a₄ = 14 and a₉ = 9. Here’s the trick: the difference between any two terms is related to the common difference and the difference in their positions in the sequence. The difference between the 9th term and the 4th term is 9 - 14 = -5. This difference of -5 happens over a span of 9 - 4 = 5 positions. So, to find 'd', we divide the difference in the terms by the difference in their positions.

Think of it this way: to go from a₄ to a₉, we take 5 steps (the difference in positions). In those 5 steps, the value decreases by 5 (the difference in the terms). So, each step (each 'd') must be -5 / 5 = -1. Therefore, the common difference, d = -1. This means that each term is 1 less than the term before it. Pretty cool, huh? Now we are one step closer to solving the whole problem, the rest is just a matter of applying our findings.

Now we know that the sequence is decreasing. It's crucial to understand how the common difference affects the sequence. If d is positive, the sequence increases; if d is negative, it decreases. In our case, since d = -1, our sequence is decreasing. This concept is fundamental in understanding the behavior of arithmetic sequences and is critical for more complex problems. Also, remember, the formula is: d = (aₙ - aₘ) / (n - m), where aₙ and aₘ are any two terms, and n and m are their positions. So, we've found our common difference, and we're ready to find the first term!

Determining the First Term (a)

Alright, we've nailed the common difference. Now, let's find the first term, 'a'. We know that a₄ = 14 and we know that d = -1. We can use the formula for the nth term of an arithmetic sequence: aₙ = a + (n - 1)d. Here, 'aₙ' is the nth term, 'a' is the first term, 'n' is the position of the term, and 'd' is the common difference. Let's use the information we have, so we can work with the fourth term, a₄ = 14. So: a₄ = a + (4 - 1)d.

We know that a₄ = 14 and d = -1. Let’s plug these values into the formula: 14 = a + (3)(-1), which simplifies to 14 = a - 3. To find 'a', we add 3 to both sides of the equation: 14 + 3 = a, so a = 17. Therefore, the first term (a₁) of the sequence is 17. We can also solve it in our heads, since the common difference is -1, and we have the 4th term, if we work our way backward, so a₃ will be 15, and a₂ will be 16, and then a₁ will be 17. Remember, knowing the first term and the common difference unlocks the entire sequence. We can now construct the whole sequence if we want: 17, 16, 15, 14, 13, 12... and so on. Keep in mind that understanding how to find the first term is essential for solving any type of arithmetic sequence problem. With 'a' and 'd' at our disposal, we can now find any term we want!

Calculating the 40th Term (a₄₀)

Now for the fun part: finding the 40th term (a₄₀). We have all the pieces of the puzzle: a = 17, d = -1, and we want to find a₄₀. Using the same formula we used before: aₙ = a + (n - 1)d. We know that n = 40 (because we want the 40th term), a = 17 (the first term), and d = -1 (the common difference). So, let's plug these values in: a₄₀ = 17 + (40 - 1)(-1). Which simplifies to: a₄₀ = 17 + (39)(-1), or a₄₀ = 17 - 39. So, a₄₀ = -22.

Boom! The 40th term in the sequence is -22. This demonstrates how understanding the formula and the common difference allows us to calculate any term, no matter how far along in the sequence it is. This is incredibly useful in various fields, like finance, where you might want to predict the value of an investment over time, or in physics, to determine the position of a particle at a specific time. Remember, the formula is your friend, and with practice, you'll be calculating terms with ease. We've gone from knowing just two terms to now being able to find the value of any term in the sequence. You're doing great!

Finding a Specific Term Number

Let’s spice things up. Suppose we need to find which term in the sequence is equal to a specific value. Let’s say, which term is -50? This means we want to find the value of 'n' where aₙ = -50. We already know a = 17 and d = -1. Using the formula aₙ = a + (n - 1)d again, we can plug in the values and solve for 'n'. So: -50 = 17 + (n - 1)(-1). First, subtract 17 from both sides: -50 - 17 = (n - 1)(-1), which simplifies to -67 = (n - 1)(-1).

Now, divide both sides by -1: -67 / -1 = n - 1, which gives us 67 = n - 1. Finally, add 1 to both sides: 67 + 1 = n, so n = 68. This means the 68th term in the sequence is -50. It’s like a reverse calculation: instead of finding the value of a term given its position, we're finding the position of a term given its value. Keep in mind that this kind of problem is very useful in any field that involves sequences and patterns. You’re becoming a sequence expert! You can calculate the nth term and also determine the position of a specific number, which means that you can analyze any type of arithmetic sequence with confidence. We've explored all the aspects of this sequence, and you've learned a lot.

Summary of Our Findings

Alright, let’s recap what we've discovered:

• Common Difference (d): -1
• First Term (a): 17
• 40th Term (a₄₀): -22
• Term Number for Value -50: 68th term

See? It wasn't that hard, right? We've successfully navigated through the problem, finding all the required elements of the arithmetic sequence. By understanding the basics and following these steps, you can tackle any arithmetic sequence problem that comes your way.

Conclusion: Your Arithmetic Sequence Adventure!

So there you have it, guys! We've successfully cracked the code of this arithmetic sequence. We started with just a few pieces of information and, step by step, uncovered all the secrets of the sequence. We found the common difference, which dictates the very nature of the sequence; the first term, which is the starting point; the 40th term, which proves that we can find any term we want; and, for good measure, the position of a specific value. I hope you found this guide helpful. Keep practicing and exploring more arithmetic sequence problems. With each problem, you'll gain more confidence and skills. Remember, math is like a muscle – the more you exercise it, the stronger it gets. Keep up the great work, and happy calculating!