Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use sigma notation to write the sum. Then use a graphing utility to find the sum. $$\left[1-\left(\frac{1}{6}\right)^{2}\right]+\left[1-\left(\frac{2}{6}\right)^{2}\right]+\cdots+\left[1-\left(\frac{6}{6}\right)^{2}\right]$$

Step-by-Step Solution

Answer

The sum in sigma notation is: \[\sum_{i=1}^{6}\left[1-\left(\frac{i}{6}\right)^{2}\right]\] To find the sum, a graphing utility is necessary.

Step 1: Identify the Sigma Notation Format

Sigma notation is a way to write a long sum in a more compact form. The format is: \[\sum_{i=a}^{b}f(i)\] where i is the index of summation, a is the lower limit, b is the upper limit, and f(i) is the function of i being summed. In our case, the function is \(1-\left(\frac{i}{6}\right)^{2}\)

Step 2: Apply Sigma Notation

The given sum starts at 1 and ends at 6, with each term in the sum corresponding to the square of the fraction with integer i from 1 to 6 divided by 6. This can be written in sigma notation as follows: \[\sum_{i=1}^{6}\left[1-\left(\frac{i}{6}\right)^{2}\right]\]

Step 3: Use a Graphing Utility to Find the Sum

At this point, use a graphing utility to calculate the sum. Simply plug in the sigma notation into the graphing utility and calculate the sum, or alternatively, sum all terms calculated individually.

Key Concepts

Summation

Summation, often represented using sigma notation \( \Sigma \), is a concise way to express adding up a sequence of numbers. Imagine you have a list where each item is a calculation based on a specific pattern or rule. Summation helps us neatly write the series that follows these patterns.
Here’s a simple breakdown:

• \( i \) is the index of summation, indicating which item you’re summing.
• \( a \) and \( b \) are the lower and upper limits, referring to the range of indices being summed.
• \( f(i) \) is the function or expression applied to each index.

For example, when expressing the sum \( 1-(\frac{1}{6})^{2} + 1-(\frac{2}{6})^{2} + \ldots + 1-(\frac{6}{6})^{2} \) in sigma notation, we write \( \sum_{i=1}^{6} [1-(\frac{i}{6})^{2}] \). This means we’re summing values derived from substituting \( i \) from 1 to 6 into the formula. Summation is essential for analyzing and evaluating mathematical sequences succinctly, helping us handle large computations easily.

Graphing Utility

A graphing utility is a powerful tool, usually in the form of a calculator or software, that can significantly aid in solving complex mathematical problems. When dealing with summations or other intricate calculations, typing everything manually can be tedious and prone to errors. Graphing utilities simplify this process.
They allow you to:

• Visualize mathematical expressions in graphs.
• Calculate the sum of complex sequences directly from inputs.
• Check your work for accuracy.

To find a sum using a graphing utility, input the formula \( \sum_{i=1}^{6}[1-(\frac{i}{6})^{2}] \) directly into the tool. It automatically computes each term’s value and finds their sum. By leveraging these technologies, you can enhance your understanding of mathematical concepts, making them more tangible and easier to grasp. If you don't have access to a physical graphing calculator, many online tools and apps provide similar capabilities for free.

Mathematical Function

A mathematical function is a relation between a set of inputs and a set of possible outputs, usually expressed with an equation. In the context of summation, the function dictates how each term in the series is computed. Functions are at the core of this process.
Let's break it down:

• The function in the given problem is \( f(i) = 1-(\frac{i}{6})^{2} \). It defines how to transform the index \( i \) into the corresponding value of the series.
• Each \( i \) represents an input from the specified range, here 1 to 6, and the function yields the value for each term in the summation.
• The consistent use of the function ensures a systematic computation for each index value.

Functions enable the concise expression of series and allow for systematic calculation and analysis of terms. Understanding how functions work in the context of a summation is crucial for both practical computations and theoretical math analyses. Moreover, exploring how a function behaves over a range makes the subject much more engaging and comprehensive.