Problem 70 - A Graphical Approach to College Algebra

Question

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. In Exercises 69 and 70 , round to the nearest thousandth. $$a_{n}=-\sqrt[3]{4} n+\sqrt{7}$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is approximately -21.611 when rounded to the nearest thousandth.

Step 1: Understand the Sequence Formula

The given arithmetic sequence formula is \( a_{n} = -\sqrt[3]{4} n + \sqrt{7} \). This means for each term, we calculate by substituting different values of \( n \), starting from \( n = 1 \) to \( n = 10 \).

Step 2: Calculate the First 10 Terms

Use the sequence formula to evaluate the first 10 terms of the sequence. This involves calculating each term individually: - \( a_1 = -\sqrt[3]{4} \cdot 1 + \sqrt{7} \) - \( a_2 = -\sqrt[3]{4} \cdot 2 + \sqrt{7} \) Continue this calculation up to \( a_{10} \). Use a calculator for precise computations.

Step 3: Sum the Terms using a Calculator

Input the sequence calculations into a graphing calculator. The calculator helps compute the values efficiently and sum them up. Ensure to check that calculator settings are correct for rounding to the nearest thousandth.

Step 4: Round the Result

After summing all 10 terms, round the result to the nearest thousandth as required by the exercise.

Key Concepts

Graphing Calculator

A graphing calculator is a trusty tool for students tackling arithmetic sequences. It allows you to enter the sequence formula and automatically calculates the terms for you. This saves you time, especially when working with complex formulas involving roots and powers like \( a_n = -\sqrt[3]{4} n + \sqrt{7} \). Start by familiarizing yourself with your specific calculator model. Explore features such as the sequence input mode, which allows you to input the initial term, common difference or multiplier, and the total number of terms you’re interested in.
When working through an arithmetic sequence, enter each term formula in succession to receive a display of the results. This lets you visualize the progression of terms and confirms you're on the right track. Once all terms are calculated, you can further use the calculator to sum these quantities quickly. Additionally, verify that all calculations stay precise by setting up your calculator to round numbers appropriately.

Sum of Sequence

An arithmetic sequence is a set of numbers ordered such that each term after the first is derived by adding a constant term, known as the common difference, to the preceding term. The sum of an arithmetic sequence refers to the addition of all its terms.
To find the sum of the first 10 terms given in the exercise, first, calculate each individual term using the given formula \( a_n = -\sqrt[3]{4} n + \sqrt{7} \). After you have calculated all 10 terms, you add them together. Using a graphing calculator can expedite this process by allowing you to directly enter the sequence formula and compute the sum automatically.
This tool efficiently handles arithmetic operations and minimizes manual calculation errors. By automating the tedious task of summing potentially complex terms, graphing calculators streamline the sequence analysis process.

Rounding Numbers

The exercise specifies rounding results to the nearest thousandth. Rounding numbers is an important skill in mathematics, ensuring solutions are presented in a clear, understandable format. This is vital in situations where precise values are not necessary or are impractical.
When rounding to the nearest thousandth, you focus on the third decimal place. For example, consider the number 123.4567. Since the digit in the fourth decimal place is 7 (greater than 5), you round up the third decimal place from 6 to 7, resulting in 123.457.
In the context of sequences, after summing the sequence on the calculator, use the rounding function to approximate your final answer. This approach not only assures consistency with exercise requirements but also helps in recognizing the significance of numerical precision.