Problem 57 - Precalculus: Mathematics for Calculus

Question

Use a graphing calculator to evaluate the sum. \(\sum_{j=7}^{20} j^{2}(1+j)\)

Step-by-Step Solution

Answer

The sum \(\sum_{j=7}^{20} j^2(1+j)\) evaluates to 82,814.

Step 1: Understand the Problem

The problem requires us to evaluate the sum \(\sum_{j=7}^{20} j^{2}(1+j)\). This is a summation from \(j = 7\) to \(j = 20\) of the expression \(j^2(1+j)\).

Step 2: Express the Term

The expression to be summed is \(j^2(1+j)\), which can be expanded to \(j^3 + j^2\). We will calculate the sum of this expanded expression from \(j = 7\) to \(j = 20\).

Step 3: Use Graphing Calculator Features

Using the summation function on a graphing calculator, input the expanded formula \(j^3 + j^2\), with the variable \(j\) ranging from 7 to 20. Most calculators have built-in methods for evaluating summations, often through menus or programming options.

Step 4: Compute the Result

Calculate the sum using the calculator by following the appropriate steps—entering the expression \(j^3 + j^2\), specifying the index variable \(j\), and setting the range from 7 to 20. The calculator performs the iterations and provides a result.

Key Concepts

Summation

Summation is a mathematical operation that involves adding together a sequence of numbers. The sum is represented by the symbol \(\sum\), which indicates that we are adding up several terms that follow a specific pattern. In the exercise example, the summation \(\sum_{j=7}^{20} j^2(1+j)\) requires calculating the sum of the expression \(j^2(1+j)\) for each integer value of \(j\) from 7 to 20.

• Simplifies complex expressions into an easier-to-understand format.
• Provides a structured way to aggregate numbers systematically.
• Makes it straightforward to use computational tools like calculators for evaluation.

Through summation, complex calculations become more manageable, especially when handled by calculators and specific mathematical functions designed for such problems. Summation is vital in various fields, including statistics, physics, and economics.

Expanded Expression

An expanded expression is the result of breaking down a mathematical expression into its simpler components. This involves applying properties of arithmetic and algebra to reveal the inner workings of the term. In the given problem, the expression \(j^2(1+j)\) is expanded to \(j^3 + j^2\). This step is crucial because:

• It simplifies the calculation by separating the components.
• Makes it easier to input into calculators, especially for repeated computations.
• Enhances the understanding of how each part contributes to the sum.

Breaking down expressions into expanded forms helps in visualizing each part of the operation and ensures accuracy when applying mathematical operations such as summation. It also aids in recognizing patterns and sequences that might not be apparent in the compact form.

Evaluate Sum

Evaluating a sum means calculating the total value of a summation expression by performing the summation itself. Using the original exercise as an example, evaluating \(\sum_{j=7}^{20} j^3 + j^2\) involves:

• Setting the variable \(j\) to start at 7 and end at 20.
• Calculating each individual term, such as \(7^3 + 7^2\) for \(j=7\) and so on, until \(j=20\).
• Using a calculator efficiently to automate these computations is key, as it speeds up the process and avoids manual entry errors.

Graphing calculators are particularly useful as they have dedicated functions to compute sums automatically. Inputting the expanded expression and the range into the calculator will output the sum almost instantaneously. Understanding how to evaluate a sum involves not only knowing the mathematical operations but also effectively using the right tools to compute them.