Problem 17 - Essential Calculus

Question

In Exercises 17 and \(18,\) use the summation capabilities of a graphing utility to evaluate the sum. Then use the properties of summation and Theorem 4.2 to verify the sum. $$ \sum_{i=1}^{20}\left(i^{2}+3\right) $$

Step-by-Step Solution

Answer

The sum of the series \( \sum_{i=1}^{20}(i^{2}+3) \) is 2930.

Step 1: Breaking the Summation

Break down the summation into two separate sums: \[ \sum_{i=1}^{20}(i^{2}+3) = \sum_{i=1}^{20}i^{2} + \sum_{i=1}^{20}3 \]. Here, we used the property of summation that allows us to distribute the sum over addition.

Step 2: Evaluate the First Summation

Evaluate the sum of squares using the formula for the sum of squares of first n natural numbers, which is \[\frac{n(n+1)(2n+1)}{6}\], where n = 20 in this case. So the sum becomes \[ \sum_{i=1}^{20}i^{2} = \frac{20(20+1)(2*20+1)}{6} = 2870 \].

Step 3: Evaluate the Second Summation

Evaluate the second summation. Here, we have 20 terms, each of which is 3, so \[ \sum_{i=1}^{20}3 = 20 * 3 = 60 \].

Step 4: Combine the Results

Combine the results obtained in Step 2 and Step 3 to get the final sum. So, the result is \[ \sum_{i=1}^{20}(i^{2}+3) = 2870 + 60 = 2930 \].

Key Concepts

Sum of Squares

The sum of squares is a concept you'll often encounter in mathematics, particularly in algebra and calculus. The sum of squares of the first \(n\) natural numbers can be calculated using a specific formula: \[\sum_{i=1}^{n} i^{2} = \frac{n(n+1)(2n+1)}{6}\]. This formula is a quick and efficient way to find the sum of squares without having to add each square individually.
For example, if you want the sum of squares for the numbers from 1 to 20, simply substitute \(n = 20\) into the formula:

• Calculate \(20 \times 21 = 420\)
• Calculate \(2 \times 20 + 1 = 41\)
• Multiply these values: \(420 \times 41 = 17220\)
• Finally, divide by 6: \(\frac{17220}{6} = 2870\)

This quick calculation gives you the sum of squares for the numbers 1 through 20.

Summation Properties

Summation is a mathematical notation used to denote the sum of a sequence of terms. It's compact and efficient. Summation has several key properties that make calculations easier:

• Linearity: The sum can be distributed over addition, meaning \(\sum(a_i + b_i) = \sum a_i + \sum b_i\).
• Constant Multiplication: A constant can be factored out of a summation, such as \(\sum(c \times a_i) = c \times \sum a_i\).
• Summation of Constants: The sum of a constant repeated \(n\) times is \(n \times c\).

These properties are useful for breaking down complex summations into manageable parts. In the given exercise, the summation is broken into two parts: \(\sum_{i=1}^{20} i^2\) and \(\sum_{i=1}^{20} 3\). This application of linearity allows us to calculate each part separately before combining them.

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical data and perform complex calculations like summation. These can be found in standalone graphing calculators or software like GeoGebra and Desmos.
Using a graphing utility, you can:

• Quickly calculate sums and visualize their graphs.
• Verify manual calculations, like the sum of squares.
• Explore mathematical patterns and properties graphically.

In our exercise, a graphing utility can be used to verify the calculated sum \(2930\). By inputting the function \(i^2 + 3\) and determining the sum of the sequence from \(i = 1\) to \(20\), the utility provides a confident check against manual calculations, saving time and reducing error.

Theorem 4.2

Theorem 4.2 is an essential concept in understanding summation and its properties. Although this particular theorem might differ slightly depending on context, it generally involves standardizing complex calculations. The theorem often provides:

• Formulas or methods to simplify computations.
• Rules on how specific functions behave under summation.
• Techniques to verify or prove results.

In our exercise, Theorem 4.2 is employed to verify the calculated sum by providing an external checkpoint. With theorem verification, you ensure the mathematical rigor of your results. This makes Theorem 4.2 invaluable for educational exercises, where accuracy and understanding form the core of learning.