Problem 13 - Essential Calculus

Question

In Exercises 13-16, use the properties of summation and Theorem 4.2 to evaluate the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{i=1}^{20} 2 i $$

Step-by-Step Solution

Answer

The sum of the series \( \sum_{i=1}^{20} 2i \) is 420.

Step 1: Identify the type of summation

The summation is of the form \( \sum_{i=1}^{n} a i \) where \( n = 20 \) and \( a = 2 \). This is a arithmetic series as it contains a fixed difference.

Step 2: Recall the formula for the sum of an arithmetic series

The formula for the sum of an arithmetic series is given by: \[ S = n/2*(a+l) \] where \( a \) is the first term, \( l \) is the last term and \( n \) is the total number of terms.

Step 3: Substitute the values into the formula

Substitute the values into the formula: \( n = 20 \), \( a = 2*1 = 2 \), \( l = 2*20 = 40 \). Therefore,\[ S = 20/2 * (2 + 40) = 10 * 42 = 420 \]

Step 4: Verify the result with a graphing calculator

Use a graphing calculator to verify the result. Input the summation from 1 to 20 for \( 2i \) into the calculator. The graphing calculator should return the result 420.

Key Concepts

Properties of Summation

Summation is a mathematical operation that adds up a sequence of numbers. It is a concise way to express the addition of many numbers using a single notation. When handling summations, especially for arithmetic series, understanding its properties is crucial. Here are some fundamental properties:

• Linearity of Summation: You can separate a summation of a sum into separate summations. If you have constants inside the summation, they can be pulled out. For example, \( \sum (a_i + b_i) = \sum a_i + \sum b_i \) and \( \sum c\cdot a_i = c \cdot \sum a_i \).
• Arithmetic Series: This is a series where the difference between consecutive terms is constant. This consistency allows for straightforward calculations using a specific formula.
• Range of Indices: The summation symbol \( \sum \) typically has a lower and an upper bound. You sum up all terms starting from the lower bound to and including the upper bound.

Understanding these properties can help simplify complex series and make calculations easier, especially when verifying calculations with tools like graphing calculators.

Summation Formula

When dealing with arithmetic series, there is a specific formula that significantly simplifies the process of finding the sum. This formula is defined as the sum of the first and last terms, divided by two, and then multiplied by the total number of terms:\[ S = \frac{n}{2} \cdot (a + l) \]where:

• \( n \): Represents the total number of terms in the series (which are counted consecutively).
• \( a \): This is the first term in the series.
• \( l \): This stands for the last term in the series.

For example, in the given exercise, our series starts at \( 2 \times 1 \) and ends at \( 2 \times 20 \).The number of terms \( n \) is 20. So, using the formula, we substituted to find that the sum \( S \) is 420. This method is not only fast but also minimizes errors as compared to adding each term individually.

Graphing Calculator Verification

After computing the arithmetic series manually, a useful method to confirm your results is through a graphing calculator. Graphing calculators have built-in functions for summation, which allows for quick verification.

• Input: First, select the summation function on your calculator.
• Set Limits: Define the lower (1) and upper (20) bounds, which specify the range of numbers you are summing.
• Expression: Input the function expression \( 2i \) , which means each term in the sum is twice the index, \( i \).
• Result: Execute the function. The calculator should yield 420 as the result, matching your manual calculations.

Utilizing a graphing calculator for verification is beneficial because it reduces human error and reinforces your understanding of the arithmetic series and summation properties through computational tools. This step is an excellent practice for ensuring the reliability and accuracy of your answers.