Sum Of First 60 Terms Arithmetic Sequence -18 -4 10 24 38 52

Hey guys! Today, we're diving into the fascinating world of arithmetic sequences. We've got a fun problem to tackle: finding the sum of the first 60 terms of the arithmetic sequence -18, -4, 10, 24, 38, 52. Don't worry, it sounds intimidating, but we'll break it down step by step so it's super easy to understand. Let's get started!

Understanding Arithmetic Sequences

Before we jump into solving the problem, let's quickly recap what arithmetic sequences are all about. Think of an arithmetic sequence as a list of numbers where the difference between any two consecutive terms is always the same. This constant difference is called the common difference.

In our sequence, -18, -4, 10, 24, 38, 52, can you spot the common difference? It's the number we add each time to get to the next term. To find it, simply subtract any term from the term that follows it. For instance, -4 - (-18) = 14, 10 - (-4) = 14, and so on. So, our common difference (often denoted as 'd') is 14. Recognizing this pattern is crucial because arithmetic sequences pop up everywhere, from simple math problems to more complex real-world scenarios, like figuring out loan payments or predicting growth patterns. Understanding the basics helps build a strong foundation for tackling more advanced stuff later on. We'll use this understanding to find not just individual terms but also the sum of a whole bunch of terms in the sequence, which is what makes this particular problem so interesting. Trust me, once you get the hang of arithmetic sequences, you'll start seeing them in all sorts of places!

Identifying Key Components

To really nail this problem, we need to identify a few key components of our arithmetic sequence. These components are like the secret ingredients in our recipe for success! First up, we have the first term, often denoted as 'a₁'. In our sequence, -18, -4, 10, 24, 38, 52, the first term (a₁) is -18. Easy peasy, right? Next, we've already talked about the common difference, 'd', which we found to be 14. This is the constant value we add to each term to get the next one. The common difference is super important because it defines the rhythm of our sequence. It’s like the beat in a song, dictating how the numbers progress. Finally, we need to know how many terms we're summing up. The problem asks us to find the sum of the first 60 terms, so the number of terms, 'n', is 60. These three little pieces of information – the first term (a₁), the common difference (d), and the number of terms (n) – are the building blocks we'll use to solve the problem. They are like the foundation of a house; without them, we can't construct our solution. So make sure you've got a good handle on these, and we'll be well on our way to cracking this sequence sum! Keep these key ingredients in mind as we move forward, and you'll see how they fit perfectly into our formula.

The Formula for the Sum of an Arithmetic Series

Alright, now for the magic formula! To find the sum of the first 'n' terms of an arithmetic sequence, we use a neat little formula. This formula is like a super-efficient shortcut that saves us from having to add up all 60 terms individually (phew!). The formula looks like this:

Sn = n/2 * [2a₁ + (n - 1)d]

Where:

• Sn is the sum of the first 'n' terms
• n is the number of terms
• a₁ is the first term
• d is the common difference

This formula might look a bit intimidating at first glance, but trust me, it's your best friend in situations like these. It's like a secret code that unlocks the answer with just a few simple substitutions. The 'n/2' part is essentially finding the average number of terms we're considering, and the '[2a₁ + (n - 1)d]' part calculates the sum of the first and last terms in a way that accounts for the common difference. Together, they give us a concise way to find the total sum. Think of it this way: instead of adding each number one by one, the formula cleverly condenses the whole process into one single calculation. It's like having a calculator that can do 60 additions in a blink! So, let's get familiar with this formula; we'll be using it a lot in these kinds of problems. Once you get the hang of it, you'll be amazed at how much time and effort it saves you. It's a total game-changer for solving arithmetic series sums!

Applying the Formula

Now comes the fun part: plugging in our values and seeing the formula in action! We know:

• n = 60 (we want the sum of the first 60 terms)
• a₁ = -18 (the first term is -18)
• d = 14 (the common difference is 14)

Let's substitute these values into our formula:

S60 = 60/2 * [2*(-18) + (60 - 1)*14]

See how we're just swapping the letters in the formula with the numbers we've identified? It's like fitting pieces into a puzzle. Now, it’s just a matter of doing the arithmetic carefully. First, let's simplify inside the brackets. We've got 2 multiplied by -18, which gives us -36. Then we have (60 - 1) which is 59, and we multiply that by 14. This is where taking it step-by-step can be super helpful to avoid making small calculation errors. Each step is a building block towards our final answer, so let's take our time and make sure each block is solid. Once we've simplified inside the brackets, the next step is to multiply by 60/2, which is simply 30. This multiplication is our final big step in the calculation, and it will reveal the sum of the first 60 terms. So, we’ve got the formula, we’ve plugged in our values, and now we're on the verge of cracking the problem wide open! Let's do the final calculations and see what the sum of the first 60 terms really is. I know you guys can do this!

Calculating the Sum

Time to crunch some numbers! Let's break down the calculation step-by-step to make sure we don't miss anything.

First, let's simplify inside the brackets:

2 * (-18) = -36

(60 - 1) * 14 = 59 * 14 = 826

Now, we add those results together:

-36 + 826 = 790

Next, we multiply by 60/2, which is 30:

S60 = 30 * 790

S60 = 23700

So, the sum of the first 60 terms of the arithmetic sequence is 23,700. Woohoo! We did it! You see, breaking down the problem into smaller, manageable steps makes even a seemingly complex calculation pretty straightforward. We started with our formula, carefully substituted the values, and then just took our time with the arithmetic. Each step led us closer to the solution, and now we've arrived at our final answer. It's like climbing a staircase, one step at a time, until you reach the top. Don't rush the process; accuracy is key. Double-check your calculations if you need to, and remember, practice makes perfect. Now that you've seen how it's done, you can tackle other arithmetic series sums with confidence. And remember, it’s not just about getting the right answer; it’s about understanding the process and building your problem-solving skills. So give yourselves a pat on the back – you’ve earned it!

Verification and Conclusion

To be absolutely sure we've nailed it, it's always a good idea to do a quick verification of our answer. While we won't manually add 60 terms (that's why we have the formula!), we can think about whether our answer seems reasonable.

The sequence is increasing, and we're adding up a lot of terms, so we expect a fairly large positive number. 23,700 seems like a plausible answer. Another way to check is to calculate the 60th term using the formula for the nth term of an arithmetic sequence: an = a₁ + (n - 1)d. Plugging in our values, we get a60 = -18 + (60 - 1) * 14 = -18 + 59 * 14 = -18 + 826 = 808. This means the 60th term is 808. Our sum of 23,700 is roughly the average of the first and last term multiplied by the number of terms, which makes sense. This quick check doesn't guarantee our answer is correct, but it gives us added confidence. Verification steps like these are invaluable in mathematics because they help us catch any silly mistakes we might have made along the way. It’s like having a safety net that ensures we don’t fall off the tightrope of problem-solving. So always take a moment to reflect on your answer and see if it fits the context of the problem. A little bit of checking can save you a lot of headaches in the long run! So, let's celebrate our success in solving this arithmetic sequence sum, and let’s keep these verification techniques in mind for all our future mathematical adventures!

In conclusion, the sum of the first 60 terms of the arithmetic sequence -18, -4, 10, 24, 38, 52 is 23,700. Great job, everyone! You've now got another tool in your math toolkit. Keep practicing, and you'll become arithmetic sequence masters in no time!