Problem 73 - 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. $$a_{n}=\sqrt{8} n+\sqrt{3}$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is approximately 172.864.

Step 1: Identify the Sequence

First, we need to identify the sequence given. The formula for the arithmetic sequence is \(a_{n} = \sqrt{8}n + \sqrt{3}\). This tells us the common difference \(d = \sqrt{8}\) and the first term \(a_1 = \sqrt{8} \cdot 1 + \sqrt{3}\).

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula to find the first term: \[ a_1 = \sqrt{8} \cdot 1 + \sqrt{3} = \sqrt{8} + \sqrt{3} \approx 2.828 + 1.732 = 4.560 \]

Step 3: Set Up the Sum Function

To evaluate the sum of the first 10 terms of the sequence, use the sum formula for an arithmetic sequence: \[ S_n = \frac{n}{2} (a_1 + a_n) \] However, since this is an arithmetic progression, we can individually sum the specific terms if needed.

Step 4: Calculate Last Term

Find the 10th term by substituting \(n = 10\) into the sequence formula: \[ a_{10} = \sqrt{8} \cdot 10 + \sqrt{3} = 10\sqrt{8} + \sqrt{3} \approx 28.284 + 1.732 = 30.016 \]

Step 5: Calculate the Sum Using a Calculator

Using a calculator, enter the sequence formula to calculate each term from \(n = 1\) to \(n = 10\), sum these calculated terms: \[ S_{10} = a_1 + a_2 + a_3 + ... + a_{10} \approx 4.560 + 7.388 + 10.216 + 13.044 + 15.872 + 18.700 + 21.528 + 24.356 + 27.184 + 30.016 = 172.864 \]

Step 6: Write the Final Answer

The sum of the first 10 terms of the sequence is approximately 172.864 rounded to the nearest thousandth.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that allows students to visualize and calculate complex mathematical functions and sequences with ease. With the built-in sequence feature, you can quickly evaluate terms in an arithmetic sequence and find their sum. Use the sequence feature to input the formula for the arithmetic sequence, such as the one described in this problem. Here, enter the formula \( a_{n} = \sqrt{8}n + \sqrt{3} \) to generate each term of the sequence. This method streamlines the calculation process by automatically computing each term from \( n = 1 \) to \( n = 10 \), which you can then sum up directly. When using a graphing calculator:

• Input the sequence formula properly.
• Check calculated terms to ensure accuracy.
• Utilize the sum function to efficiently add all terms in the specified range.

Lastly, remember to round the calculated sum to the nearest thousandth if required. This ensures precision and clarity in your final answer.

Sum of Sequence

The sum of a sequence refers to adding up all the individual terms in a specific range of an arithmetic sequence. For an arithmetic sequence, this is often done using a sum formula. In this exercise, the sequence is given by \( a_{n} = \sqrt{8}n + \sqrt{3} \), which is identified as being arithmetic, hence having a constant difference between its terms. There are two common methods to find the sum:

• Manual Summation: Calculate each term, such as \( a_1, a_2, \ldots, a_{10} \), and add them together manually.
• Using the sum formula: \( S_n = \frac{n}{2} (a_1 + a_n) \). This requires calculating only the first and last terms, and then computing the sum with these values.

Both methods aim to simplify the process of finding the total of the sequence over a specific range. For accurate results, especially with complex sequences, a graphing calculator is incredibly helpful to automate and verify manual calculations.

Arithmetic Progression

An arithmetic progression (or arithmetic sequence) is a sequence of numbers in which the difference between any two consecutive terms is constant. This fixed difference is known as the common difference, denoted by \( d \). For instance, given the sequence formula \( a_{n} = \sqrt{8}n + \sqrt{3} \), the common difference here is \( \sqrt{8} \). Key characteristics of an arithmetic progression include:

• The common difference \( d \) is the same throughout the sequence.
• You can easily calculate any term using the formula for the nth term \( a_n = a_1 + (n-1) \, d \).
• Such sequences are represented as straight lines when graphed, due to the linear nature of their formulas.

Understanding arithmetic progressions is essential, as it forms the basis for many algebraic concepts and problem-solving methods. Recognizing patterns and formulas in arithmetic progressions allows for efficient computations in mathematical solutions.