First Negative Term: Sequence 750, 734, 718, 702...

Hey guys! Ever stumbled upon a sequence of numbers and wondered when it's going to dip into the negatives? Well, today we're diving deep into just that! We've got a sequence here: 750, 734, 718, 702, and our mission, should we choose to accept it, is to pinpoint the very first negative term. Sounds like a mathematical adventure, right? Let's get started!

Cracking the Code: Understanding Arithmetic Sequences

Before we jump into solving this specific problem, let's quickly recap what an arithmetic sequence actually is. Arithmetic sequences are sequences where the difference between consecutive terms is constant. This constant difference is often called the 'common difference'. Think of it like climbing a staircase where each step is the same height. You start at a certain level, and each step adds (or subtracts) the same amount to your height. That consistent change? That’s your common difference!

In our sequence (750, 734, 718, 702...), we can easily spot that the numbers are decreasing. To find the common difference, we simply subtract any term from the term that follows it. So, 734 - 750 = -16. Similarly, 718 - 734 = -16, and 702 - 718 = -16. Aha! The common difference (d) is -16. This means that each subsequent term is 16 less than the previous one. Understanding this consistent pattern is crucial because it allows us to predict future terms in the sequence. We now know that this isn't just a random collection of numbers; it's a structured descent towards negative territory!

Knowing the common difference is like having a key piece of the puzzle. It tells us the rate at which the sequence is changing. A negative common difference, as we have here, indicates a decreasing sequence. A positive common difference would, of course, mean the sequence is increasing. This foundational understanding of arithmetic sequences sets us up perfectly to tackle the challenge of finding that elusive first negative term. Now that we know the sequence is decreasing by 16 each time, we can start thinking about how many steps it will take to go from 750 all the way down to a negative number. We're essentially trying to figure out how many times we need to subtract 16 before we cross the zero threshold. This brings us to the next important step: formulating a general expression for the nth term of the sequence. This formula will be our secret weapon in pinpointing exactly when the sequence turns negative.

The Nth Term Formula: Our Secret Weapon

Now that we've identified our sequence as arithmetic and found the common difference, it's time to introduce the star of the show: the nth term formula. This formula is the magic key that unlocks any term in the sequence, no matter how far down the line it is. The general formula for the nth term (an) of an arithmetic sequence is:

an = a1 + (n - 1)d

Where:

• an is the nth term (the term we want to find)
• a1 is the first term of the sequence
• n is the term number (the position of the term in the sequence)
• d is the common difference

Let's break this down and see how it applies to our specific problem. In our sequence (750, 734, 718, 702...), we already know: a1 = 750 (the first term) and d = -16 (the common difference). So, we can plug these values into our formula to get a specific expression for this sequence:

an = 750 + (n - 1)(-16)

This formula now gives us a way to calculate any term in the sequence just by plugging in the term number (n). For example, if we wanted to find the 10th term, we'd substitute n = 10 into the formula. But our goal is a bit different. We don't want to find a specific term; we want to find the first term that's negative. This means we're looking for the smallest value of 'n' that makes 'an' less than zero. Think of it like setting a target. Our target is zero, and we want to find the first time the sequence dips below that target. This is where our formula really shines. It allows us to turn our problem into an inequality, which we can then solve for 'n'.

The power of this formula lies in its ability to connect the term number ('n') directly to the value of the term ('an'). It bridges the gap between position and value within the sequence. Without this formula, we'd be stuck manually calculating each term until we stumbled upon a negative one, which could take a very long time. But with this formula, we have a systematic way to find our answer. It's like having a GPS for our sequence, guiding us directly to our destination: the first negative term. Now, let's put this formula to work and find that term!

Setting the Stage: When Does the Sequence Turn Negative?

Okay, guys, we've got our nth term formula (an = 750 + (n - 1)(-16)), and we know we're looking for the first negative term. This means we need to find the smallest whole number 'n' for which 'an' is less than zero. In mathematical terms, we need to solve the following inequality:

750 + (n - 1)(-16) < 0

This inequality is the heart of our problem. It translates our question – “When does the sequence turn negative?” – into a concrete mathematical statement. Think of it like setting the rules of our game. We're saying, “Find the smallest 'n' that makes this statement true.” To solve this inequality, we'll need to use our algebraic skills. We'll simplify the expression, isolate 'n', and then interpret the result. It's like decoding a secret message! Each step in the algebraic process will bring us closer to revealing the value of 'n' that marks the transition from positive to negative terms in our sequence.

Before we dive into the algebra, let's take a moment to appreciate what we're doing here. We're not just plugging numbers into a formula; we're using the power of mathematics to predict the future behavior of the sequence. We're taking a pattern and extrapolating it beyond the given terms. This is a fundamental aspect of mathematical thinking – the ability to generalize and make predictions based on established relationships. The beauty of this approach is that it's not specific to this particular sequence. The same principles can be applied to any arithmetic sequence, allowing us to find the first negative term (or any other specific term) with ease. This makes our method a powerful and versatile tool in the world of sequences and series. So, let's roll up our sleeves and get this inequality solved! We're on the verge of finding our answer, and the algebraic steps we're about to take are the final steps in our journey.

The Grand Finale: Solving the Inequality and Finding the Term

Alright, let's get down to the nitty-gritty and solve that inequality! We have:

750 + (n - 1)(-16) < 0

First, we distribute the -16:

750 - 16n + 16 < 0

Next, combine the constants:

766 - 16n < 0

Now, let's isolate the term with 'n'. We can do this by subtracting 766 from both sides:

-16n < -766

And now, the crucial step: dividing both sides by -16. Remember, when we divide (or multiply) an inequality by a negative number, we need to flip the inequality sign! So:

n > -766 / -16

n > 47.875

So, what does this tell us? It tells us that 'n' must be greater than 47.875 for the term to be negative. But 'n' represents the term number, and it has to be a whole number (we can't have a 47.875th term!). Therefore, we need to round up to the next whole number. The smallest whole number greater than 47.875 is 48. So, n = 48. This means the 48th term is the first negative term in the sequence!

But we're not quite done yet! We've found the position of the first negative term, but we haven't found the value of the term itself. To do this, we simply plug n = 48 back into our nth term formula:

a48 = 750 + (48 - 1)(-16)

a48 = 750 + (47)(-16)

a48 = 750 - 752

a48 = -2

Boom! There it is. The first negative term in the sequence is -2. We've cracked the code! We started with a sequence of numbers, used our understanding of arithmetic sequences and the nth term formula, and navigated our way to the first negative term. This is a fantastic example of how mathematical tools can be used to solve real-world problems (or, in this case, sequence-world problems!).

Wrapping Up and Celebrating Our Victory

So, there you have it, guys! We successfully navigated the world of arithmetic sequences and pinpointed the first negative term in the sequence 750, 734, 718, 702... It turned out to be the 48th term, with a value of -2. We achieved this by understanding the concept of common difference, wielding the powerful nth term formula, and fearlessly tackling an inequality. This journey highlights the beauty and practicality of mathematics. It's not just about memorizing formulas; it's about understanding the underlying principles and applying them to solve problems.

We started by defining arithmetic sequences and understanding the crucial role of the common difference. Then, we introduced the nth term formula, our secret weapon for finding any term in the sequence. We used this formula to set up an inequality, which allowed us to determine when the sequence would turn negative. Finally, we solved the inequality, found the term number, and calculated the value of the first negative term. Each step built upon the previous one, demonstrating the logical and sequential nature of mathematical problem-solving.

This process isn't just about finding the answer; it's about developing problem-solving skills that can be applied to a wide range of situations. The ability to identify patterns, formulate equations, and solve them systematically is a valuable asset in any field. So, the next time you encounter a sequence of numbers, remember the tools and techniques we've discussed today. You might just surprise yourself with what you can discover! And remember, mathematics is not just about numbers and equations; it's about logic, reasoning, and the thrill of solving a puzzle. We hope you enjoyed this mathematical adventure as much as we did. Until next time, keep exploring the fascinating world of numbers!