Calculate & Round: Σ_{n=0}^{54} 70(1.12)^{n+1}
Hey math enthusiasts! Today, we're diving into a fun little problem that involves calculating the sum of a series and rounding the result to the nearest integer. Specifically, we're looking at the expression: Σ_{n=0}^{54} 70(1.12)^{n+1}. Don't worry, it looks a bit intimidating at first glance, but we'll break it down step by step, so even if you're not a math whiz, you'll be able to follow along. This is all about understanding the components, applying the right formula, and then getting that rounded answer. So, grab your calculators (or your preferred calculation method), and let's get started! We will explore this question thoroughly in this article. We will break down the formula and then use a calculator. Finally, we will round the result to the nearest integer.
Understanding the Expression Σ_{n=0}^{54} 70(1.12)^{n+1}
Alright, let's unpack this mathematical beast. First, what does Σ mean? It's the summation symbol, telling us to add up a series of terms. In this case, we're adding terms together as n goes from 0 to 54. Inside the summation, we have 70(1.12)^n+1}. This is the general term for each value of n. Let's break down the general term for each value and add them all up. This is a geometric series because each term is obtained by multiplying the previous term by a constant factor (in this case, 1.12). Geometric series pop up everywhere, from compound interest calculations to modeling population growth, so understanding them is super useful.
To make it even clearer, let's write out the first few terms of this series to show you guys what's going on. When n = 0, the term is 70(1.12)^{0+1} = 70(1.12)^1 = 78.4. When n = 1, the term is 70(1.12)^{1+1} = 70(1.12)^2 = 87.808. When n = 2, the term is 70(1.12)^{2+1} = 70(1.12)^3 = 98.345. See how each term is multiplied by 1.12 compared to the previous one? That's the characteristic of a geometric series. As n continues to increase up to 54, we're calculating similar values and adding them all together. The key here is recognizing that it's a geometric series. This allows us to use a handy formula to calculate the sum without having to manually add up all 55 terms (from n=0 to n=54). The formula for the sum of a finite geometric series is a real lifesaver when you encounter these types of problems. Using this, we will find the sum of this expression.
Applying the Geometric Series Formula
Okay, now for the good part – the formula! For a geometric series, the sum (S) of the first n terms is given by:
S = a(1 - r^n) / (1 - r)
Where:
- a is the first term of the series.
- r is the common ratio (the factor by which each term is multiplied).
- n is the number of terms.
In our case:
- a = 70(1.12)^{0+1} = 78.4 (the first term when n = 0)
- r = 1.12 (the common ratio)
- n = 55 (since n goes from 0 to 54, there are 55 terms)
Now, let's plug these values into the formula:
S = 78.4 * (1 - 1.12^{55}) / (1 - 1.12)
Be careful with the order of operations here. Make sure you calculate the exponent (1.12^{55}) first, then subtract it from 1, and so on. Let's calculate the value of 1.12^{55}. This equals approximately 1,079.317. Then the formula will look like this S = 78.4 * (1 - 1,079.317) / (1 - 1.12). We will go over this and solve it step by step, so that it is easy to understand. Using the calculator is very useful, and will make sure our calculations are right.
Let's continue to the next step. So, in the previous step we had S = 78.4 * (1 - 1,079.317) / (1 - 1.12). We will calculate the values in the parentheses first. So we have, (1 - 1,079.317) = -1,078.317. The next step will be, (1 - 1.12) = -0.12. Then the formula will look like this, S = 78.4 * (-1,078.317) / (-0.12). We can proceed by calculating the numerator first. 78.4 * (-1,078.317) = -84,529.564. Then the formula looks like S = -84,529.564 / (-0.12). Dividing both values will give us the result. -84,529.564 / (-0.12) = 704,413.033. This is the sum of the series. Then, we can move on to the next step, where we round to the nearest integer. This will conclude our math exercise.
Rounding to the Nearest Integer
We've calculated the sum! Now, all that's left is to round the result to the nearest integer. Our calculated sum is approximately 704,413.033. When rounding to the nearest integer, we look at the decimal part. If it's 0.5 or greater, we round up; otherwise, we round down. In our case, the decimal part is 0.033, which is less than 0.5. Therefore, we round down. So, rounding 704,413.033 to the nearest integer gives us 704,413. This is our final answer! See, wasn't that bad, guys? We started with a seemingly complex expression, but by breaking it down, applying the right formula, and taking it step by step, we arrived at a solution. The key takeaways here are:
- Recognizing the geometric series.
- Knowing the formula for the sum of a geometric series.
- Carefully applying the formula.
- Remembering how to round to the nearest integer.
Understanding these concepts is really helpful for a wide range of math problems. You will have to do this in many cases. Whether it's compound interest or any other type of math problems, these formulas and understanding of series will be helpful. Keep practicing, and you'll get better and better at these types of problems.
Final Answer and Conclusion
So, to recap, the value of the expression Σ_{n=0}^{54} 70(1.12)^{n+1}, rounded to the nearest integer, is 704,413. We hope you found this explanation helpful and easy to follow. Remember, the beauty of mathematics lies in its ability to break down complex problems into manageable steps. If you have any more math problems, feel free to share them! Keep exploring, keep questioning, and most importantly, keep enjoying the world of numbers! The important part is to remember the formula. Then you will easily be able to calculate and determine the results of the problems.
Thanks for joining me on this math adventure, and happy calculating!