/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 49 Use a graphing calculator to eva... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 49

Use a graphing calculator to evaluate each series. $$\sum_{j=3}^{9}\left(3 j-j^{2}\right)$$

Short Answer

Expert verified
The sum of the series \(ewline \sum_{j=3}^{9}(3j-j^{2})\) is \(\ -77\).

Step by step solution

01

Understand the series

The series to evaluate is \(ewline \sum_{j=3}^{9}(3j-j^{2})\). This means summing the expression \(3j-j^{2}\) from \(j=3\) to \(j=9\).
02

Input the series into the graphing calculator

Turn on the graphing calculator and access the summation function. Enter \(ewline \sum_{j=3}^{9}(3j-j^{2})\) into the calculator.
03

Evaluate the series

After entering the series, the calculator will compute the sum of the expression \(3j-j^{2}\) from \(j=3\) to \(j=9\).
04

Write down the result

Record the result provided by the graphing calculator. This is the value of the series.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Calculator
A graphing calculator is a powerful tool used for mathematical computations like evaluating series. It provides functions to compute complex sums easily. In our exercise, we need to use a graphing calculator to evaluate the series \(\textstyle \sum_{j=3}^{9}(3j-j^{2})\). To start, make sure your graphing calculator is turned on. Navigate to the summation function, typically found under the math menu. Input the series expression and set the bounds for the summation (from j=3 to j=9). The calculator will process the input and display the result.
Summation Notation
Summation notation (using the Greek letter sigma \(\textstyle \sum\)) is a concise way to represent the sum of a sequence of terms. The notation \(\textstyle \sum_{j=3}^{9}(3j-j^{2})\) tells you the following: you start summing when j=3 and continue up to j=9. For each value of j within this range, you calculate \(\textstyle 3j-j^{2}\) and then add these values together. This method is essential for working with sequences and series in mathematics, helping to structure and simplify the computation steps.
Series Computation
Series computation involves summing the terms of a sequence according to a specific rule. In our case, \(\textstyle 3j-j^{2}\) is the rule applied for each j from 3 to 9. Here鈥檚 how you can break it down:
  • Plug in j=3: \(\textstyle 3 \times 3 - 3^{2} = 9 - 9 = 0\)
  • Plug in j=4: \(\textstyle 3 \times 4 - 4^{2} = 12 - 16 = -4\)
  • Continue this pattern for j=5, j=6, and so on, up to j=9
Once you have all these values, you sum them to get the total value of the series. By using a graphing calculator, the steps are simplified and the final result is computed quickly without manual calculations.
Step-by-Step Problem Solving
Step-by-step problem solving is crucial for understanding and correctly performing series evaluations:
  • Step 1: Understand the Series - Identify the starting point, ending point, and the expression within the summation.
  • Step 2: Input into Calculator - Make use of the summation function in your graphing calculator and input the series expression correctly.
  • Step 3: Evaluate the Series - Allow the graphing calculator to process and compute the total sum of the series.
  • Step 4: Write Down the Result - Record the result displayed on the calculator. This is the solution to your series.
By following these steps, you ensure clarity and accuracy in solving series problems, making the process manageable and less prone to errors.

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