Summation Calculation A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon a mathematical expression that looks like a cryptic code? Well, today, we're going to crack one of those codes together. We're diving into the fascinating world of summation, specifically, how to calculate the sum of a series. Our mission, should we choose to accept it, is to find the sum of the series represented by the expression $\sum_{n=5}^8(3n^2 + 27)$. Don't worry if it looks intimidating now; by the end of this guide, you'll be a summation whiz!

Cracking the Summation Code: Understanding the Basics

Before we jump into the nitty-gritty calculations, let's break down what this expression actually means. The symbol '$\sum$' is the Greek capital letter sigma, and in mathematics, it's the universal symbol for summation. Think of it as a mathematical instruction to add up a series of terms. Now, let's dissect the other parts of our expression:

• The Index Variable (n): This is our counter, the variable that changes with each term in the series. In our case, 'n' is like a little explorer, starting its journey at 5 and ending at 8.
• The Lower Limit (5): This tells us where our index variable begins its adventure. We start our calculations with n = 5.
• The Upper Limit (8): This is the destination for our index variable. We stop our calculations when n reaches 8.
• The Expression Inside the Summation (3n² + 27): This is the rule, the formula that determines each term in our series. We'll plug in different values of 'n' into this expression to generate the terms we need to add up.

So, in a nutshell, $\sum_{n=5}^8(3n^2 + 27)$ is asking us to do the following:

1. Start with n = 5.
2. Plug n = 5 into the expression 3n² + 27 to get the first term.
3. Increase n by 1 (so n becomes 6).
4. Plug n = 6 into the expression to get the second term.
5. Keep doing this until n reaches 8.
6. Finally, add up all the terms we've calculated.

It's like following a recipe, but instead of ingredients, we have numbers and a formula. Once we understand this fundamental concept, the rest is just arithmetic! So, let's roll up our sleeves and start calculating.

The Summation Journey Begins: Calculating Each Term

Alright, guys, now that we've deciphered the summation code, it's time to get our hands dirty with some actual calculations. We'll systematically plug in each value of 'n' from 5 to 8 into our expression, 3n² + 27, and see what terms we get.

Term 1: When n = 5

Let's start with the first stop on our summation journey, n = 5. We substitute 5 for 'n' in our expression:

3(5)² + 27 = 3(25) + 27 = 75 + 27 = 102

So, the first term in our series is a solid 102. We're off to a great start!

Term 2: When n = 6

Next up, we move along the summation path to n = 6. We plug in 6 for 'n' in our expression:

3(6)² + 27 = 3(36) + 27 = 108 + 27 = 135

Our second term is 135. Notice how the terms are getting larger as 'n' increases. This is because of the n² term in our expression, which makes the value grow quadratically.

Term 3: When n = 7

We're halfway there! Now, let's calculate the term when n = 7:

3(7)² + 27 = 3(49) + 27 = 147 + 27 = 174

The third term is 174. The numbers are climbing steadily, adding to the suspense of the final sum!

Term 4: When n = 8

Finally, we reach the last stop on our summation journey, n = 8. We plug in 8 for 'n' in our expression:

3(8)² + 27 = 3(64) + 27 = 192 + 27 = 219

Our final term is 219. We've successfully navigated the expression for each value of 'n', and now we have all the pieces of the puzzle.

So, to recap, we've calculated the following terms:

• When n = 5, the term is 102.
• When n = 6, the term is 135.
• When n = 7, the term is 174.
• When n = 8, the term is 219.

Now comes the exciting part: adding these terms together to find the total sum! Get ready to see our summation efforts come to fruition.

The Grand Finale: Summing Up the Terms

We've reached the final leg of our journey, the moment we've all been waiting for: adding up the terms we so diligently calculated. Remember, the summation symbol '$\sum$' is just a fancy way of telling us to add things together. So, let's bring those terms together and see what grand total we arrive at.

We have the following terms:

• 102 (when n = 5)
• 135 (when n = 6)
• 174 (when n = 7)
• 219 (when n = 8)

Now, let's add them up:

102 + 135 + 174 + 219 = ?

Grab your calculators, or put on your mental math hats, and let's do this! Adding the numbers together, we get:

102 + 135 = 237

237 + 174 = 411

411 + 219 = 630

Eureka! We've found it! The sum of the series $\sum_{n=5}^8(3n^2 + 27)$ is 630. We've successfully navigated the summation expression, calculated each term, and added them all up to reveal the final answer.

The Summation Symphony: A Recap of Our Journey

Wow, what a journey we've had! We started with a seemingly complex expression, $\sum_{n=5}^8(3n^2 + 27)$, and through careful dissection and calculation, we've arrived at the solution: 630. Let's take a moment to recap the key steps we took:

1. Understanding the Basics: We deciphered the summation notation, identifying the index variable, lower and upper limits, and the expression to be summed.
2. Calculating Each Term: We systematically plugged in each value of 'n' from 5 to 8 into the expression 3n² + 27, generating the individual terms of the series.
3. Summing Up the Terms: We added all the calculated terms together to find the total sum.

Throughout this process, we've not only solved a specific problem but also gained a deeper understanding of the concept of summation. We've seen how a seemingly complex expression can be broken down into manageable steps, and how careful calculation can lead us to a precise answer.

Summation Superpowers: Unleashing Your Mathematical Potential

Now that you've mastered this summation challenge, you're equipped with a powerful tool for tackling other mathematical problems. Summation is a fundamental concept in mathematics, appearing in various fields like calculus, statistics, and computer science. By understanding how to calculate sums, you're unlocking a whole new level of mathematical potential.

So, go forth and explore the world of summations! Try different expressions, experiment with different limits, and challenge yourself to find the sums. The more you practice, the more confident and proficient you'll become. And remember, math isn't just about numbers and formulas; it's about problem-solving, critical thinking, and the joy of discovery. Keep exploring, keep learning, and keep unleashing your mathematical superpowers!

Summation Calculation A Step-by-Step Guide

Find the sum of the series (3n^2 + 27) where n ranges from 5 to 8. Show each step of your work.