Calculating Teresa's Computer Purchase Total Cost In 15 Installments

Hey guys! Let's dive into Teresa's computer purchase and break down the financials, this falls squarely into accounting, and we will explore how to calculate the total cost of her new machine given her payment plan. Teresa decided to buy a computer and opted for a payment plan of 15 monthly installments. The initial installment is R$ 200, and each subsequent month, the installment amount increases by R$ 10. The core challenge here lies in determining the total amount Teresa will pay for the computer. This situation presents a classic arithmetic progression problem, where we have a series of numbers (the monthly payments) that increase by a constant amount (R$ 10). To solve this, we need to understand the principles of arithmetic sequences and how to apply them to real-world financial scenarios. We’ll look at the step-by-step calculation to find the total cost, making sure we cover all the essential elements of this financial puzzle. Whether you’re a student, a finance enthusiast, or just curious, understanding these calculations can be super useful. So, let’s get started and figure out how much Teresa will ultimately pay for her computer!

To really get a grip on how much Teresa is paying, we need to break down the payment plan step by step. Let's talk about the structure of Teresa’s payment plan in detail. She's paying in 15 monthly installments, which means we have a series of payments to consider. The first payment kicks off at R$ 200, that’s our starting point. Now, here's the interesting part: each month, the payment goes up by R$ 10. This consistent increase is what makes this an arithmetic progression. Understanding this pattern is crucial because it allows us to predict the payment amount for any given month. For instance, the second month's payment will be R$ 210 (R$ 200 + R$ 10), the third will be R$ 220 (R$ 210 + R$ 10), and so on. This steady increase is a key characteristic of arithmetic sequences. To calculate the total amount paid, we need to sum up all these individual payments over the 15 months. This isn't as simple as multiplying the initial payment by the number of months because the payments aren't constant. Instead, we need to use the formula for the sum of an arithmetic series, which we’ll dive into later. By understanding the payment structure—the initial amount and the consistent increase—we can set the stage for calculating the total cost of the computer. This methodical approach not only helps us solve this specific problem but also equips us with the tools to tackle similar financial scenarios. So, let’s keep this payment structure in mind as we move forward and explore the mathematical tools we’ll need to find the final answer.

Now, let's get to the nitty-gritty of calculating the total cost. To figure out how much Teresa will pay in total, we need to use the formula for the sum of an arithmetic series. This formula is super handy when you have a series of numbers that increase (or decrease) by a constant amount, just like Teresa's monthly payments. Here’s the formula we’ll be using: $S_n = \fracn}{2} [2a + (n - 1)d]$ Where - Sn is the sum of the first n terms (the total amount paid). - n is the number of terms (in this case, 15 months). - a is the first term (the initial payment of R$ 200). - d is the common difference (the increase of R$ 10 each month). Let’s break it down and plug in the numbers. We know that n is 15, a is 200, and d is 10. So, the formula becomes: $S_{15 = \frac15}{2} [2(200) + (15 - 1)10]$ Now, let's simplify step by step First, calculate the values inside the brackets: $2(200) = 400$ $(15 - 1)10 = 14 imes 10 = 140$ Next, add these results together: $400 + 140 = 540$ Now, plug this back into the formula: $S_{15 = \frac15}{2} imes 540$ Multiply and divide $S_{15 = 7.5 imes 540$ $S_{15} = 4050$ So, the total amount Teresa will pay for the computer is R$ 4050. This calculation shows how powerful the arithmetic series formula can be for solving real-world financial problems. By understanding and applying this formula, we’ve been able to determine the total cost of Teresa’s computer. Let's move on and summarize our findings in a clear and concise conclusion.

Alright, guys, let's wrap things up! We've journeyed through Teresa's computer purchase and crunched the numbers to find out the total cost. After laying out the payment structure and applying the arithmetic progression formula, we’ve arrived at a solid conclusion. Teresa will pay a total of R$ 4050 for her computer. This final amount gives us a clear picture of Teresa’s financial commitment over the 15-month payment period. By understanding the principles of arithmetic sequences, we were able to accurately calculate the total cost, even with the increasing monthly payments. This exercise highlights the importance of financial planning and understanding how different payment structures can impact the overall cost. Whether it’s a computer, a car, or any other big purchase, knowing how to calculate the total cost can help you make informed decisions. So, the next time you encounter a similar situation, remember the steps we took: break down the payment structure, identify the pattern, and apply the appropriate formula. And there you have it! Teresa’s computer will cost her R$ 4050 in total. We hope this detailed breakdown has been helpful and has given you some valuable insights into financial calculations. Keep these tools in your financial toolkit, and you’ll be well-prepared to tackle similar scenarios in the future.

{