Problem 68 - 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. $$a_{n}=8.42 n+36.18$$

Step-by-Step Solution

Answer

The sum of the first 10 terms is 824.9.

Step 1: Identify the Arithmetic Sequence Formula

The sequence is given by the formula \( a_n = 8.42n + 36.18 \). This tells us how each term in the sequence is calculated depending on its position \( n \).

Step 2: Determine the First 10 Terms

Use the formula \( a_n = 8.42n + 36.18 \) to calculate the first 10 terms of the sequence by substituting \( n = 1 \), \( n = 2 \), and so on up to \( n = 10 \).

Step 3: Calculate the Terms

Evaluate each term by substituting the values:- \( a_1 = 8.42(1) + 36.18 = 44.6 \)- \( a_2 = 8.42(2) + 36.18 = 53.02 \)- Continue this calculation until \( a_{10} = 8.42(10) + 36.18 = 120.38 \).

Step 4: Sum the First 10 Terms

Add all the calculated terms: \( 44.6 + 53.02 + 61.44 + 69.86 + 78.28 + 86.7 + 95.12 + 103.54 + 111.96 + 120.38 \).

Step 5: Evaluate the Sum

Complete the addition: \( 44.6 + 53.02 + 61.44 + 69.86 + 78.28 + 86.7 + 95.12 + 103.54 + 111.96 + 120.38 = 824.9 \).

Step 6: Round if Necessary

Since the result is \( 824.9 \) and we need to round to the nearest thousandth, the sum remains as it is: \( 824.9 \).

Key Concepts

Sum of Terms in an Arithmetic Sequence

In any arithmetic sequence, the sum of a certain number of terms can be calculated using either a formula or by adding individual terms.
For our example, the first 10 terms were directly added. Calculating them individually gives a clear picture of the sequence.
However, you can also use the sum formula for arithmetic sequences:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

Here, \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the last term. In our case, \( n = 10 \), \( a_1 = 44.6 \), and \( a_{10} = 120.38 \). This method is efficient, especially for longer sequences.

Exploring Sequences with a Graphing Calculator

A graphing calculator is a powerful tool for visualizing and solving problems related to sequences.
It not only simplifies the arithmetic calculations but also provides visual insights. Here are some steps to use a graphing calculator with sequences:

• Enter the sequence formula: In most graphing calculators, you can input the general expression of the sequence, like \( a_n = 8.42n + 36.18 \).
• Calculate specific terms: You can ask your calculator to substitute values of \( n \) to get particular terms.
• Sum functions: Check if your calculator offers functions to sum the sequence terms automatically. This can save you time and reduce error margins.

This technological aid ensures accurate results and a better understanding of sequences by visualizing them.

Understanding the Sequence Formula

The sequence formula defines how each term in the sequence is derived. In our example, the formula \( a_n = 8.42n + 36.18 \) helps us generate terms based on their position \( n \).
Here's a breakdown:

• \( 8.42 \): The coefficient of \( n \) is known as the common difference. It indicates how much each term increases over the previous one.
• \( 36.18 \): This is the starting value, or the first term when \( n = 0 \), in an adjusted sequence.

By substituting different values of \( n \), you can generate as many terms as needed. This sequence formula is linear, creating a straight-line pattern if plotted.

Evaluation of Arithmetic Sequences

Evaluating arithmetic sequences involves determining specific terms and their sum. This is often achieved through calculations based on the sequence formula.
When solving a sequence:

• Identify each term by replacing \( n \) in the formula with sequential numbers.
• Sum the terms for a total value if needed.

The manual approach provides an understanding of the underlying arithmetic operations involved. For our sequence, each term was calculated and then summed to get \( 824.9 \), demonstrating a hands-on approach to sequence evaluation.