Arithmetic Sequence Example Finding Common Difference Of -3

Hey guys! Ever get those math questions that seem like they're speaking another language? Today, we're diving deep into arithmetic sequences, and we're going to break down one of those tricky questions together. We'll explore what an arithmetic sequence is, how to identify one, and then nail that question about the common difference of -3. Let's make math a bit less intimidating and a lot more fun!

Understanding Arithmetic Sequences

Okay, so what exactly is an arithmetic sequence? Simply put, an arithmetic sequence is a series of numbers where the difference between any two consecutive terms is always the same. This constant difference is called the common difference. Think of it like climbing stairs where each step is the same height.

To really grasp this, let's break it down further. Imagine we have a sequence: 2, 4, 6, 8, 10. What's happening here? We're adding 2 to each term to get the next one. That '2' is our common difference. It’s the magic number that keeps the sequence consistent. Now, let’s say we have another sequence: 1, 5, 9, 13, 17. Here, we're adding 4 each time, so our common difference is 4. See the pattern? It’s all about that consistent addition (or subtraction, as we'll see).

But what if we encounter a sequence like 1, 3, 7, 15, 31? Notice anything different? We're not adding the same number each time. The difference between 1 and 3 is 2, but the difference between 3 and 7 is 4. This sequence isn't arithmetic because it doesn't have a common difference. It's a different kind of sequence altogether, maybe geometric or something else. The key takeaway here is that consistency is king in arithmetic sequences. If you spot a consistent difference, you've got yourself an arithmetic sequence. And that common difference? That’s your golden ticket to understanding the sequence’s behavior and predicting its future terms. So, keep an eye out for that constant change, and you'll be an arithmetic sequence pro in no time!

Identifying the Common Difference

Now that we know what an arithmetic sequence is, let's talk about how to pinpoint that all-important common difference. The common difference, as we've discussed, is the constant value added (or subtracted) to get from one term to the next in the sequence. Finding it is actually pretty straightforward. The formula we use is super simple: subtract any term from the term that follows it. That's it!

Let's put this into action with an example. Suppose we have the sequence: 5, 10, 15, 20, 25. To find the common difference, we can pick any two consecutive terms. Let's take 10 and 5. We subtract the first term (5) from the second term (10): 10 - 5 = 5. So, our common difference is 5. Easy peasy, right? We could have also chosen 20 and 15: 20 - 15 = 5. Same answer! This works because, by definition, the common difference is consistent throughout the sequence. If you get a different difference when you subtract different pairs of terms, then guess what? It's not an arithmetic sequence.

But what if the sequence is decreasing? No problem! The same principle applies. Let’s look at the sequence: 30, 25, 20, 15, 10. If we subtract 30 from 25, we get 25 - 30 = -5. Notice the negative sign? This tells us that the common difference is -5, meaning we're subtracting 5 each time. So, the common difference can absolutely be negative, indicating that the sequence is decreasing. This is super important to remember! Common differences aren't limited to positive numbers; they can be negative or even zero (in which case the sequence would be constant, like 7, 7, 7, 7...). So, always pay attention to the sign when calculating the common difference. It gives you crucial information about the sequence's trend – whether it's going up, going down, or staying put.

Analyzing the Given Sequences

Alright, now we're armed with the knowledge to tackle the question at hand. We need to identify which sequence has a common difference of -3. Remember, this means we're looking for a sequence where each term is 3 less than the term before it. Let's dive into the options one by one and see what we find.

A. 3, -9, 27, -81, 243: At first glance, this sequence looks like it's changing a lot, but let’s check the differences. If we subtract 3 from -9, we get -9 - 3 = -12. Okay, that's already not -3. But just to be sure, let's check another pair. If we subtract -9 from 27, we get 27 - (-9) = 36. Definitely not an arithmetic sequence with a common difference of -3. This sequence is actually a geometric sequence (where you multiply by a constant factor), but that’s a topic for another time. For now, we can confidently say this isn't our answer.

B. 24, 21, 18, 15, 5: This sequence looks promising because it's decreasing. Let's calculate the differences. 21 - 24 = -3. So far, so good! Let's check the next pair: 18 - 21 = -3. Awesome! And one more, just to be absolutely sure: 15 - 18 = -3. But wait a second! What happens when we look at the last two terms? 5 - 15 = -10. Uh oh! The common difference isn't consistent throughout the sequence. It was -3 for the first few terms, but then it changed. This means this sequence is not arithmetic, and it's not our answer either. Tricky, right? You've got to check every pair of terms to be certain.

C. 43, 40, 37, 34, 31: This sequence also appears to be decreasing, which is what we'd expect with a negative common difference. Let's calculate: 40 - 43 = -3. Perfect start! Next, 37 - 40 = -3. Looking good! 34 - 37 = -3. Still on track! And finally, 31 - 34 = -3. Bingo! The common difference is consistently -3 throughout the entire sequence. This is our winner! We've found an arithmetic sequence with a common difference of -3. High five!

D. 144, 48, 18, 6, 2: This sequence is definitely decreasing, but the changes look pretty dramatic. Let's calculate the differences: 48 - 144 = -96. Whoa, that's not -3! No need to check further; this isn't an arithmetic sequence with a common difference of -3. This sequence is another geometric sequence (we're dividing by a factor each time), but we're not focusing on those right now.

The Correct Sequence

Drumroll, please! After carefully analyzing each sequence, we've found our answer. Sequence C. 43, 40, 37, 34, 31 is the arithmetic sequence with a common difference of -3. We figured this out by calculating the difference between consecutive terms and confirming that it was consistently -3 throughout the entire sequence.

So, what did we learn today? We dove deep into arithmetic sequences, understood what a common difference is, and mastered the technique of identifying it. We also learned the importance of checking all the terms in a sequence to make sure the common difference is consistent. Math problems like these can seem daunting at first, but by breaking them down step by step and applying the right concepts, we can conquer them. Keep practicing, keep exploring, and remember, math can be fun!

Arithmetic Sequence Example Finding Common Difference of -3

Which sequence is an arithmetic sequence with a common difference of d = -3?