Finding The 12th Term Of An Arithmetic Sequence

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Hey guys! Ever stumbled upon a math problem that looks intimidating at first glance? Well, arithmetic sequences might seem like one of those, but trust me, they're super manageable once you understand the basics. Today, we're going to break down a classic arithmetic sequence problem: finding the 12th term of a sequence where the first term is 4 and the common difference is 5. Sounds interesting? Let's dive in!

Understanding Arithmetic Sequences

Before we jump into solving the problem, let's make sure we're all on the same page about what an arithmetic sequence actually is. An arithmetic sequence is simply a list of numbers where the difference between any two consecutive terms is constant. This constant difference is what we call the common difference. Think of it like climbing stairs where each step is the same height. The height of each step is your common difference.

  • The First Term (a): This is the starting point of our sequence. It's the number you kick things off with. In our problem, the first term, often denoted as a, is 4. So, we start our sequence with the number 4. It's like the foundation upon which the rest of the sequence is built.
  • The Common Difference (b): The common difference, typically represented by b, is the consistent amount we add (or subtract) to get from one term to the next. In this case, our common difference is 5. This means we add 5 to each term to get the subsequent term. It’s the rhythm of the sequence, the beat that keeps it going.
  • The Nth Term (an): Now, this is where it gets interesting. The nth term, written as an, is the term we want to find at a specific position (n) in the sequence. For example, if we want to find the 5th term, n would be 5. Our mission today is to find the 12th term, so n is 12. This is the treasure we're hunting for!

To summarize, an arithmetic sequence is all about identifying these key components: the first term, the common difference, and the term you're trying to locate. Once you've got these, you're well on your way to mastering arithmetic sequences.

The Formula for the Nth Term

Now that we've got a solid grasp of what arithmetic sequences are, let's talk about the magic formula that will help us find any term in the sequence without having to list them all out. This formula is the key to unlocking the value of any term, no matter how far down the line it is. Ready to see it?

The formula for finding the nth term (an) of an arithmetic sequence is:

an = a + (n - 1) * b

Let's break this down piece by piece so we understand exactly what's going on:

  • an: This is what we're trying to find – the nth term. It's our target, the value we're aiming to calculate. In our problem, this is the 12th term.
  • a: This is the first term of the sequence, the starting number. We already know this from our problem; it's 4.
  • n: This represents the position of the term we want to find. If we're looking for the 12th term, n is 12. It tells us where in the sequence we're looking.
  • b: This is the common difference, the constant value we add to each term to get the next. In our problem, b is 5.

So, in plain English, this formula tells us: "To find the nth term, take the first term, add the common difference multiplied by one less than the term number." See? Not so scary when we break it down. This formula is a powerful tool, allowing us to jump directly to any term in the sequence without having to calculate all the preceding terms. It's like having a shortcut on a long journey!

Applying the Formula to Our Problem

Alright, guys, now for the exciting part! We've got the formula, we've got the pieces of our puzzle, and now we're going to put them together to solve for the 12th term of our arithmetic sequence. Remember, our first term (a) is 4, the common difference (b) is 5, and we want to find the 12th term, which means n is 12. Let's plug these values into our formula:

a12 = a + (12 - 1) * b

Now, let's substitute the values we know:

a12 = 4 + (12 - 1) * 5

Time for some good old-fashioned arithmetic! First, we'll tackle the parentheses:

a12 = 4 + (11) * 5

Next up, multiplication:

a12 = 4 + 55

And finally, the grand finale – addition:

a12 = 59

There you have it! The 12th term of the arithmetic sequence is 59. We did it! By plugging in our known values and following the order of operations, we've successfully navigated the formula and found our answer. This process highlights the power of the formula; it transforms a potentially tedious task into a straightforward calculation. So, whenever you encounter an arithmetic sequence problem, remember this method, and you'll be solving them like a pro.

Step-by-Step Solution

Sometimes, seeing the solution laid out step-by-step can make things even clearer. So, let's recap the process we just went through, breaking it down into individual steps. This is like having a detailed roadmap for solving arithmetic sequence problems. Ready to trace our steps?

  1. Identify the given values:

    • First term (a) = 4
    • Common difference (b) = 5
    • Term number (n) = 12 (since we want to find the 12th term)

    This first step is crucial. It's about extracting the information we already have from the problem. Think of it as gathering your ingredients before you start cooking. Knowing what you have on hand sets the stage for the rest of the solution.

  2. Write down the formula for the nth term:

    an = a + (n - 1) * b

    This formula is our trusty tool. It's the blueprint we'll use to construct our solution. Writing it down reminds us of the relationship between the terms and helps us stay organized.

  3. Substitute the values into the formula:

    a12 = 4 + (12 - 1) * 5

    Here, we're plugging in the values we identified in step one. It's like fitting the pieces of a puzzle into the correct spots. Accuracy is key here; make sure each value goes where it belongs.

  4. Simplify the expression using the order of operations (PEMDAS/BODMAS):

    • First, solve the expression inside the parentheses:

      a12 = 4 + (11) * 5

    • Next, perform the multiplication:

      a12 = 4 + 55

    • Finally, do the addition:

      a12 = 59

    This step is all about careful calculation. Following the order of operations ensures we get the correct answer. It's like following the instructions in a recipe to make sure the dish turns out perfectly.

  5. State the answer:

    The 12th term of the arithmetic sequence is 59.

    This is where we declare our victory! We've found the solution, and now we're stating it clearly. It's like presenting the finished dish after all our hard work in the kitchen.

By breaking the problem down into these steps, we've made it much less daunting. Each step is a manageable task, and together, they lead us to the solution. Remember, problem-solving is often about breaking things down into smaller, more digestible parts. So, next time you're faced with a similar problem, remember these steps, and you'll be well-equipped to tackle it.

Conclusion

So, guys, we've successfully navigated the world of arithmetic sequences and found the 12th term of our sequence. We started by understanding what arithmetic sequences are, then we learned the formula for finding the nth term, and finally, we applied that formula to our specific problem. We even broke down the solution into step-by-step instructions to make it super clear.

Remember, math problems might seem tough at first, but with a little understanding and the right tools, you can conquer them. Keep practicing, keep exploring, and you'll be amazed at what you can achieve. And who knows, maybe you'll even start to enjoy the challenge! Until next time, keep those math skills sharp!