Finding The 27th Term Arithmetic Sequence A_1=38 And A_17=-74

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Hey guys! Ever wondered how to pinpoint a specific term in a sequence of numbers? Today, we're diving deep into the world of arithmetic sequences to uncover the mystery behind finding the 27th term. We've got some initial clues: the first term ($a_1$) is 38, and the 17th term ($a_{17}$) is -74. Buckle up, because we're about to embark on a mathematical adventure!

Understanding Arithmetic Sequences

First things first, let's make sure we're all on the same page about what an arithmetic sequence actually is. Simply put, it's a sequence where the difference between any two consecutive terms is constant. This constant difference is what we call the common difference, often denoted by 'd'. Think of it like climbing stairs where each step is the same height. That consistent step height is your common difference.

To really nail this down, consider a few examples. The sequence 2, 4, 6, 8... is an arithmetic sequence because we're adding 2 each time. The common difference here is 2. Similarly, 10, 7, 4, 1... is also an arithmetic sequence, but this time we're subtracting 3, making the common difference -3. Spotting these patterns is key to cracking arithmetic sequences.

Now, why is this concept so important? Well, arithmetic sequences pop up everywhere in real life, from simple patterns to complex calculations. Imagine stacking chairs where each row has a fixed number more than the row before, or calculating interest earned over time with consistent deposits. Understanding these sequences gives us a powerful tool for prediction and problem-solving in a multitude of situations. So, with a solid grasp of what arithmetic sequences are, we're ready to tackle our main challenge: finding the 27th term.

The Formula for Success: Unveiling the General Term

To conquer our challenge, we need a trusty weapon: the formula for the general term of an arithmetic sequence. This formula is our secret sauce, allowing us to calculate any term in the sequence without having to manually list them all out. The formula looks like this:

an=a1+(n−1)da_n = a_1 + (n - 1)d

Let's break this down, shall we? $a_n$ is the term we're trying to find (in our case, $a_{27}$). $a_1$ is the first term of the sequence, which we know is 38. 'n' is the position of the term we want (27th in our problem). And 'd' is that crucial common difference we talked about earlier. The beauty of this formula is that if we know $a_1$ and 'd', we can find any term in the sequence just by plugging in the right 'n'.

But here's the catch: we don't yet know 'd'! That's the missing piece of our puzzle. We've got $a_1$ and we know $a_{17}$, but to use our powerful formula to find $a_{27}$, we need to figure out 'd' first. Think of it like needing the right key to unlock a door. The formula is the door, and 'd' is the key. So, how do we find this elusive common difference? That's what we'll tackle next.

Cracking the Code: Finding the Common Difference

Alright, let's put on our detective hats and hunt down that common difference, 'd'. Remember, we know $a_1 = 38$ and $a_{17} = -74$. We can use the same general term formula, but this time, we'll plug in what we know about $a_{17}$ to create an equation. This equation will be our breadcrumb trail leading us to 'd'.

So, let's substitute n = 17 into our formula: $a_{17} = a_1 + (17 - 1)d$. Now, we can replace $a_{17}$ with -74 and $a_1$ with 38: $-74 = 38 + 16d$. See what we've done? We've transformed our problem into a simple algebraic equation! Now, it's just a matter of solving for 'd'.

First, we need to isolate the term with 'd'. Let's subtract 38 from both sides of the equation: $-74 - 38 = 16d$, which simplifies to $-112 = 16d$. Now, the final step! To get 'd' by itself, we divide both sides by 16: $d = -112 / 16$, which gives us $d = -7$. Eureka! We've found our key. The common difference is -7. This means each term in our sequence is 7 less than the term before it. Now that we have 'd', we're ready to use our formula to find the 27th term.

The Grand Finale: Calculating the 27th Term

With the common difference (d = -7) in our grasp, we're finally ready to find the 27th term ($a_{27}$). This is the moment we've been building up to! We'll use the general term formula one last time, plugging in all the values we now know: $a_1 = 38$, n = 27, and d = -7.

Let's revisit the formula: $a_n = a_1 + (n - 1)d$. Substituting our values, we get: $a_27} = 38 + (27 - 1)(-7)$. Time to simplify! First, we calculate the expression inside the parentheses $27 - 1 = 26$. So now we have: $a_{27 = 38 + 26(-7)$. Next, we multiply 26 by -7: $26 * -7 = -182$. Our equation now looks like this: $a_{27} = 38 - 182$. Finally, we subtract 182 from 38: $38 - 182 = -144$.

And there we have it! The 27th term of our arithmetic sequence is -144. We've successfully navigated the sequence and pinpointed our target term. Guys, give yourselves a pat on the back – you've conquered this mathematical challenge!

Wrapping Up: Key Takeaways and Why This Matters

So, what have we learned on this mathematical journey? We've not only found the 27th term of a specific arithmetic sequence, but we've also reinforced some fundamental concepts. We started by understanding what arithmetic sequences are, recognizing that constant common difference is the key. We then armed ourselves with the general term formula, a powerful tool for finding any term in a sequence. We tackled the crucial step of finding the common difference, using the information we had about $a_1$ and $a_{17}$. Finally, we put it all together to calculate $a_{27}$.

But why does this matter? Arithmetic sequences, as we touched on earlier, are more than just abstract math concepts. They are the building blocks for understanding patterns and making predictions in various real-world scenarios. From finance (calculating simple interest) to physics (analyzing motion with constant acceleration) to computer science (understanding linear data structures), the principles of arithmetic sequences are at play. By mastering these concepts, you're not just learning math; you're developing critical thinking and problem-solving skills that will serve you well in many areas of life.

So, next time you encounter a pattern, remember our adventure today. Think about the common difference, the general term formula, and the power of arithmetic sequences. You might just surprise yourself with what you can discover! Keep exploring, keep questioning, and most importantly, keep having fun with math!