Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth. $$a_{n}=-\sqrt[3]{4} n+\sqrt{7}$$

Step-by-Step Solution

Answer

Use a calculator to find and sum the first 10 terms, rounding the total to the nearest thousandth.

Step 1: Identify the Formula for the Arithmetic Sequence

The given sequence formula is \(a_n = -\sqrt[3]{4} n + \sqrt{7}\). This is an explicit formula for the arithmetic sequence where \(a_n\) represents the nth term.

Step 2: Use the Formula to Calculate the First 10 Terms

Use the sequence formula to calculate each term from \(n=1\) to \(n=10\). Substitute each value of \(n\) into the formula: \(-\sqrt[3]{4} n + \sqrt{7}\). You will calculate each term individually.

Step 3: Calculate Each Term

Calculate each of the first 10 terms methodically:- For \(n=1\), \(a_1 = -\sqrt[3]{4} \times 1 + \sqrt{7}\).- Continue calculating similarly up to \(n=10\), using a calculator to find exact values and rounding each to the nearest thousandth.

Step 4: Find the Sum of the First 10 Terms

Sum the values you have calculated for \(a_1, a_2, \ldots, a_{10}\). With these 10 calculated terms, add them together to find the total sum.

Step 5: Round to the Nearest Thousandth

Once you have the sum from Step 4, round this sum to the nearest thousandth to get the final result.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that can help visualize sequences, among many other mathematical functions. For arithmetic sequences like the one given in the exercise, it simplifies the process of evaluating terms and their sums.

• Inputting Formulas: You can input the explicit formula directly into the calculator. For the sequence \(a_{n} = -\sqrt[3]{4} n + \sqrt{7}\), the calculator will help generate each term by substituting \(n\) values.
• Visual Representation: While it can be used for calculations, a graphing calculator can also plot the terms on a graph, providing a visual insight into how the sequence behaves as \(n\) increases.
• Calculating Sums: Many graphing calculators have functions specifically designed to calculate the sum of terms of a sequence automatically, saving time and reducing errors from manual calculation.

This ability to both calculate and visualize makes graphing calculators invaluable for understanding and solving sequences.

Sum of Sequence

The sum of a sequence, especially in arithmetic sequences, involves adding together several terms that follow a defined pattern. In our given sequence, the task was to find the sum of the first 10 terms:

• Determine the Terms: Each term \(a_n\) is calculated using the given explicit formula, \(a_{n} = -\sqrt[3]{4} n + \sqrt{7}\), for \(n\) ranging from 1 to 10.
• Perform the Addition: Once all the terms are calculated, they must be summed up to find the total. This can be manually intensive but straightforward using a calculator.
• Rounding the Sum: As often required, once the sum is obtained, it should be rounded to the nearest thousandth for precision and clarity in reporting.

Understanding the concept of summing sequences gives insight into how patterns or trends evolve over specified terms.

Explicit Formula

An explicit formula provides a direct way to find any term in a sequence without needing to know the previous term, which is useful for arithmetic sequences. In the given exercise, the formula is\[ a_{n} = -\sqrt[3]{4} n + \sqrt{7} \]

• Arithmetic Sequence Type: Unlike recursive formulas, explicit ones do not require computation of preceding terms. Simply plug in the value of \(n\) to find \(a_n\).
• Analysis of Formula: The term \(-\sqrt[3]{4} n\) signifies a linear decrease as \(n\) increases, while \(\sqrt{7}\) remains a constant part of every term. This shows the sequential nature of arithmetic sequences.
• Practical Application: You can easily use the explicit formula to predict the value of any specific term or find a series of terms quickly without extensive computation.

Mastering explicit formulas grants significant advantages in both efficiency and understanding of sequences.