Calculating S90: Arithmetic Sequence Explained

Hey Plastik Magazine readers! Ever find yourselves scratching your heads over arithmetic sequences? Don't worry, you're not alone! Today, we're diving deep into how to calculate the sum of the first 90 terms, denoted as $S_{90}$, for a specific arithmetic sequence. We'll break it down step by step so that even if math isn't your forte, you'll walk away feeling like a pro. Let's get started, shall we?

Understanding Arithmetic Sequences

Before we jump into the calculation, let's make sure we're all on the same page about what an arithmetic sequence actually is. An arithmetic sequence is essentially a list of numbers where the difference between any two consecutive terms is constant. This constant difference is what we call the common difference, often represented by the letter 'd'. Think of it like a staircase where each step is the same height – that constant height is your 'd'.

In our case, the arithmetic sequence is defined by the formula ${a_n} = 71 - 2.3n$. This formula tells us how to find any term in the sequence. The 'n' represents the term number (1st, 2nd, 3rd, and so on), and plugging it into the formula gives us the value of that term. For example, if we want to find the first term ($a_1$), we substitute n = 1 into the formula: $a_1 = 71 - 2.3(1) = 68.7$. Similarly, the second term ($a_2$) would be $71 - 2.3(2) = 66.4$, and so forth. By understanding this fundamental concept, we set the stage for calculating the sum of the first 90 terms with confidence.

The common difference, 'd', is crucial for identifying and working with arithmetic sequences. To find 'd' in our sequence, we simply subtract any term from the term that follows it. So, $d = a_2 - a_1 = 66.4 - 68.7 = -2.3$. This means that each term in the sequence is 2.3 less than the term before it. Recognizing this pattern is key to understanding the behavior of the sequence and ultimately, to calculating $S_{90}$. Once we grasp the concept of the common difference and how it dictates the sequence's progression, we can confidently move forward to the next stage: determining the formula we'll use to calculate the sum of the first 90 terms.

The Formula for the Sum of an Arithmetic Sequence

Now that we have a solid understanding of arithmetic sequences, let's talk about how to actually calculate the sum of a certain number of terms. There's a handy formula that makes this process much easier than adding up all the terms individually. The formula for the sum of the first 'n' terms of an arithmetic sequence, denoted as $S_n$, is:

S_n = rac{n}{2}(a_1 + a_n)

Where:

• $S_n$ is the sum of the first 'n' terms.
• 'n' is the number of terms we want to sum.
• $a_1$ is the first term in the sequence.
• $a_n$ is the nth term in the sequence.

This formula might look a little intimidating at first, but let's break it down. The 'n/2' part is simply half the number of terms we're summing. The (a_1 + a_n) part is the sum of the first term and the nth term. So, essentially, the formula is saying that the sum of the first 'n' terms is equal to half the number of terms multiplied by the sum of the first and last terms. This is a super efficient way to calculate the sum, especially when dealing with a large number of terms like 90!

To use this formula effectively, we need to know three things: the number of terms ('n'), the first term ($a_1$), and the nth term ($a_n$). In our specific problem, we want to find $S_{90}$, so 'n' is 90. We've already calculated $a_1$ in the previous section. Now, all that's left to do is figure out what $a_{90}$ is. Once we have all these pieces, we can plug them into the formula and get our answer. So, let's move on to calculating the 90th term, and then we'll be ready to put everything together.

Calculating the 90th Term ($a_{90}$)

Alright, guys, to use our sum formula, we need to know the 90th term ($a_{90}$) of the sequence. Remember our formula for the sequence: ${a_n} = 71 - 2.3n$? This formula is our ticket to finding any term we want!

To find $a_{90}$, we simply substitute n = 90 into the formula. This gives us:

$a_{90} = 71 - 2.3(90)$

Now, let's do the math. First, we multiply 2.3 by 90, which equals 207. Then, we subtract that from 71:

$a_{90} = 71 - 207 = -136$

So, the 90th term in our sequence is -136. That's a pretty significant drop from our first term, 68.7, which makes sense given that our common difference is -2.3. This calculation is a crucial step because $a_{90}$ is a key ingredient in our sum formula. Without it, we wouldn't be able to find $S_{90}$. Now that we have $a_{90}$, along with 'n' and $a_1$, we're finally ready to put everything together and calculate the sum of the first 90 terms. Are you excited? Because I am! Let's get to the final calculation!

With the 90th term in hand, we're now equipped to tackle the final calculation. This step brings together all the individual components we've worked on so far, illustrating how each piece contributes to the overall solution. Understanding the value of the 90th term provides a clearer picture of the sequence's progression, especially when dealing with a relatively large number of terms. Knowing $a_{90}$ not only allows us to use the sum formula effectively but also deepens our understanding of the sequence's behavior. This comprehensive approach ensures that we're not just plugging numbers into a formula, but truly grasping the underlying concepts.

Calculating $S_{90}$

Okay, folks, the moment we've been waiting for! We have all the pieces we need to calculate $S_{90}$, the sum of the first 90 terms of our arithmetic sequence. Let's bring back our formula:

S_n = rac{n}{2}(a_1 + a_n)

We know:

• n = 90
• $a_1$ = 68.7
• $a_{90}$ = -136

Now, let's plug these values into the formula:

S_{90} = rac{90}{2}(68.7 + (-136))

First, we simplify the fraction: 90/2 = 45. Then, we add 68.7 and -136, which gives us -67.3. Now we have:

$S_{90} = 45(-67.3)$

Finally, we multiply 45 by -67.3, which gives us:

$S_{90} = -3028.5$

So, there you have it! The sum of the first 90 terms of the arithmetic sequence defined by ${a_n} = 71 - 2.3n$ is -3028.5. We did it! We took a potentially daunting problem and broke it down into manageable steps. This calculation demonstrates the power of understanding formulas and applying them systematically. By correctly identifying the components and substituting them into the equation, we arrived at the final answer with confidence.

The final result, $S_{90} = -3028.5$, not only provides the numerical solution but also offers insights into the overall behavior of the sequence. The negative sum suggests that the negative terms in the sequence have a greater cumulative impact than the positive terms within the first 90 terms. This reinforces the importance of considering the common difference and its influence on the sequence's progression. Furthermore, by understanding the mechanics of this calculation, we empower ourselves to tackle similar problems in the future, solidifying our grasp of arithmetic sequences and their applications. This entire process showcases how mathematical problems, when approached methodically, can be demystified and solved effectively.

Conclusion

Awesome job, guys! We've successfully calculated $S_{90}$ for our arithmetic sequence. We started by understanding what an arithmetic sequence is, then we learned the formula for calculating the sum of its terms, and finally, we put it all together to find our answer. Remember, the key to tackling math problems is breaking them down into smaller, more manageable steps. Don't be afraid to revisit the basics and understand the underlying concepts. With a little practice, you'll be solving these problems like a piece of cake! Keep exploring, keep learning, and stay tuned for more math adventures here at Plastik Magazine!

By revisiting the basics and understanding the underlying concepts, we build a strong foundation for tackling more complex mathematical challenges. The ability to break down a problem into smaller, manageable steps is a valuable skill that extends beyond mathematics, applicable in various areas of life. This exercise not only provides a numerical answer but also reinforces the importance of persistence and a methodical approach to problem-solving. Keep exploring different mathematical concepts, and remember that practice is the key to mastery. The more you engage with these ideas, the more confident and proficient you'll become in navigating the world of mathematics and beyond. So, keep pushing your boundaries, and never stop learning!