Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{i=4}^{9} 3(0.25)^{i}$$

Step-by-Step Solution

Answer

The sum is approximately 0.004.

Step 1: Understand the Problem

The problem asks us to evaluate the sum of a series from 4 to 9 of the expression \(3 \times (0.25)^i\). We will use a graphing calculator to find this value and then round the result to the nearest thousandth.

Step 2: Calculate Each Term in the Series

Before calculating, check each term in the series: - For \(i = 4\), compute \(3 \times (0.25)^4 = 3 \times 0.00390625\).- For \(i = 5\), compute \(3 \times (0.25)^5 = 3 \times 0.0009765625\).- For \(i = 6\), compute \(3 \times (0.25)^6 = 3 \times 0.000244140625\).- For \(i = 7\), compute \(3 \times (0.25)^7 = 3 \times 0.00006103515625\).- For \(i = 8\), compute \(3 \times (0.25)^8 = 3 \times 0.0000152587890625\).- For \(i = 9\), compute \(3 \times (0.25)^9 = 3 \times 0.000003814697265625\).

Step 3: Sum All Terms

Add together all the calculated terms: \[3 \times 0.00390625 + 3 \times 0.0009765625 + 3 \times 0.000244140625 + 3 \times 0.00006103515625 + 3 \times 0.0000152587890625 + 3 \times 0.000003814697265625.\] Use a calculator to obtain the sum.

Step 4: Use the Graphing Calculator

Enter the expression into the graphing calculator by using the summation notation function. Input: \( \sum_{i=4}^{9} 3 \times (0.25)^i \). The calculator will output the sum.

Step 5: Round the Result

Take the calculated sum from the calculator and round it to the nearest thousandth. Make sure you follow the rounding rules where if the fourth decimal place is 5 or more, you round up.

Key Concepts

Series Evaluation

When approaching a series evaluation problem, the goal is to find the total sum of a set of values, each calculated based on a given formula. In this exercise, we are summing terms defined by the expression \(3(0.25)^i\) for \(i\) from 4 to 9.

To do this manually:

• First, compute each term by substituting values of \(i\) from 4 to 9 into the expression \(3(0.25)^i\).
• Next, add all these computed terms to obtain the sum.

Utilizing a graphing calculator simplifies this task, since you can leverage built-in functions to automate evaluations and summations.

This process is helpful when dealing with geometric series or when the series terms exhibit exponential behavior, as repeatedly multiplying can be time-consuming without technology. The graphing calculator can accurately handle decimal precision, which is essential for rounding accurately to the nearest thousandth, as required here.

Summation Notation

Summation notation, denoted by the Greek letter Sigma (\(\sum\)), is a concise way to represent the sum of terms over a specified index set, often integers. In our example, we express the series from \(i = 4\) to \(i = 9\) for the function \(3(0.25)^i\).

In this notation:

• The variable \(i\) is known as the index of the summation.
• The lower bound (4) is where \(i\) begins, and the upper bound (9) is where it ends.
• The expression \(3(0.25)^i\) is what is evaluated for each integer value of \(i\).

Using a graphing calculator, you can input this notation directly to find the total sum. This method provides accuracy and saves time compared to manually calculating each term.

Mastery of summation notation is crucial for solving problems in calculus and statistics, as it lays the groundwork for understanding series and integration.

Rounding to the Nearest Thousandth

Rounding is a key skill in mathematics that allows us to approximate numbers to a desired degree of accuracy. For this exercise, we need to round the result of the summation to the nearest thousandth, which means rounding to three decimal places.

Here's how rounding works:

• Look at the digit in the fourth decimal place after your calculated number.
• If this digit is 5 or greater, round the third decimal place up by one.
• If the digit is less than 5, keep the third decimal place the same.

This principle ensures the result is as accurate as needed for further calculations or making conclusions. It's particularly important for results obtained from calculators, where precision is required.

Having a clear understanding of rounding rules helps maintain consistency in scientific notation, statistics, and real-life applications like finance and engineering.