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In mathematics, summation notation is a way to represent the sum of a sequence of terms. It is denoted by the symbol \( \Sigma \). In the original exercise, we see this in action with \(\sum_{i=1}^{10}(4i^2 - 5)\). The \( \Sigma \) symbol tells us to sum the expression \(4i^2 - 5\) as \(i\) goes from 1 to 10. This means:

• When \(i = 1\), the term is \(4(1)^2 - 5\)
• When \(i = 2\), the term is \(4(2)^2 - 5\)
• ... and so on until \(i = 10\).

The result is the sum of all these individual terms. This notation is powerful because it simplifies the representation of large sums and series, making it easier to work with complex calculations.

Graphing calculators are powerful tools that can do much more than just plot graphs. They also have functions that allow you to evaluate series and perform summations quickly. For the given exercise:

• First, identify the summation function, usually marked by \( \Sigma \).
• Set the lower limit of the summation (i.e., \(i = 1\)).
• Set the upper limit (i.e., \(i = 10\)).
• Enter the function to be summed, which is \( 4i^2 - 5 \).

Once these values are entered, you simply press a button to compute the sum. The graphing calculator saves you significant time and reduces errors by automating the sum's evaluation.

Evaluating a series means finding the sum of a sequence of terms. In the exercise, the series is given by \( \sum_{i=1}^{10}(4i^2 - 5) \). To evaluate this:
1. Identify the formula or function inside the summation symbol (here, \( 4i^2 - 5 \)).
2. Determine the range of values for the summation variable (\(i\) varies from 1 to 10).
3. Manually calculate each term (\( 4(1)^2 - 5 \) for \(i = 1\)), (\( 4(2)^2 - 5 \) for \(i = 2\)), etc.
4. Add up all those terms to get the final sum.

Using a graphing calculator simplifies this process. After entering the function and the range into the calculator, it performs all the tedious calculations and gives you the result instantly. This makes it much easier to handle large sums and reduces the possibility of making mistakes.