Summation Series Calculation And Verification A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of summation! Specifically, we're going to tackle the problem of finding the sum of a series by adding each term individually. But, that's not all! We'll also learn how to use the powerful summation capabilities of a graphing utility to double-check our results. Think of it as having a super-smart calculator buddy to make sure we're on the right track. So, buckle up and let's get started!

Understanding Summation Notation

Before we jump into the calculations, let's make sure we're all on the same page when it comes to summation notation. You know, those fancy symbols that look like a stretched-out Greek letter E (Σ)? This symbol, called sigma, is the key to expressing the sum of a series in a concise way. So, let's break it down, shall we?

At its core, summation notation provides a compact way to represent the sum of a sequence of numbers. Instead of writing out each term individually and adding them up (which can be tedious for long sequences), we use the sigma notation to express the sum in a more elegant and efficient manner. The general form of the summation notation looks like this:

∑[expression] from i = [start_value] to [end_value]

Let's dissect each part of this expression:

• Σ (Sigma): This is the summation symbol, indicating that we're going to sum up a series of terms.
• expression: This is the formula or rule that defines each term in the series. It usually involves a variable, often denoted as 'i' or 'k', which represents the index of the term.
• i = [start_value]: This specifies the starting value of the index variable (e.g., i = 1). The summation begins with the term corresponding to this index value.
• [end_value]: This indicates the ending value of the index variable (e.g., 5). The summation continues until we reach the term corresponding to this index value.

So, in essence, the summation notation tells us to take the expression, plug in each value of the index variable from the start value to the end value, and then add up all the resulting terms. Now that we've got the basics down, let's see how this applies to our specific problem.

Problem: $\sum_{i=1}^5(7 i-5)$

Okay, now we get to the fun part – tackling our specific problem! We're given the summation: $\sum_{i=1}^5(7 i-5)$. Let's break this down and understand what it's asking us to do. Remember our dissection of the summation notation? Time to put that knowledge to use!

In this summation: $\sum_{i=1}^5(7 i-5)$:

• The summation symbol (Σ) tells us we're dealing with a sum.
• The expression (7i - 5) defines how each term in the series is calculated. This is where the variable i comes into play. It's the index that changes with each term.
• The lower limit, i = 1, tells us that the summation starts with the first term, where i is equal to 1.
• The upper limit, 5, indicates that the summation ends with the fifth term, where i is equal to 5.

So, what does this all mean? It means we need to do the following:

1. Substitute i = 1 into the expression (7i - 5) to find the first term.
2. Substitute i = 2 into the expression (7i - 5) to find the second term.
3. Continue this process for i = 3, i = 4, and i = 5 to find the remaining terms.
4. Finally, add up all the terms we've calculated to get the final sum.

Sounds like a plan? Let's get those calculations rolling!

Calculating the Sum by Adding Each Term

Alright, let's roll up our sleeves and crunch some numbers! We're going to calculate the sum by plugging in each value of i into the expression (7i - 5) and then adding up the results. Remember, we need to do this for i = 1, 2, 3, 4, and 5. Let's take it one step at a time to ensure we don't miss anything.

Step 1: i = 1

Substitute i = 1 into the expression (7i - 5):

(7 * 1) - 5 = 7 - 5 = 2

So, the first term in our series is 2. Easy peasy, right?

Step 2: i = 2

Now, let's plug in i = 2:

(7 * 2) - 5 = 14 - 5 = 9

The second term is 9. We're on a roll!

Step 3: i = 3

Let's keep going with i = 3:

(7 * 3) - 5 = 21 - 5 = 16

The third term is 16. Getting the hang of this?

Step 4: i = 4

Time for i = 4:

(7 * 4) - 5 = 28 - 5 = 23

The fourth term is 23. Almost there!

Step 5: i = 5

Last but not least, let's plug in i = 5:

(7 * 5) - 5 = 35 - 5 = 30

The fifth term is 30. We've got all our terms!

Now that we've calculated each term individually, the final step is to add them all together. So, let's add 2 + 9 + 16 + 23 + 30. What do we get? 80! Woohoo! We've found the sum by adding each term together. But, remember, we're not done yet. We need to verify our result using a graphing utility. Let's jump into that next!

Verifying the Result with a Graphing Utility

Okay, team, we've done the hard work of calculating the sum by hand. But, it's always a good idea to double-check our work, especially in mathematics! That's where our trusty graphing utility comes in. These tools are super handy for verifying summations and other calculations. So, let's see how we can use one to confirm our result.

Most graphing calculators and online graphing tools have built-in summation functions. These functions allow you to directly input the summation expression and get the result. The exact steps might vary slightly depending on the specific calculator or software you're using, but the general idea is the same.

Here's a general outline of how you'd typically use a graphing utility to verify a summation:

1. Access the Summation Function: Look for a button or menu option that represents summation. It might be labeled with the sigma symbol (Σ),