Problem 12 - A Graphical Approach to College Algebra

Question

Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=3-\sqrt{2}, a_{2}=3$$

Step-by-Step Solution

Answer

3 - \sqrt{2}, 3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}.

Step 1: Identify the first term

The first term of the sequence, denoted as \(a_1\), is given as \(3 - \sqrt{2}\).

Step 2: Determine the second term

The second term of the sequence, denoted as \(a_2\), is given as \(3\).

Step 3: Calculate the common difference

To find the common difference \(d\), subtract the first term \(a_1\) from the second term \(a_2\): \[ d = a_2 - a_1 = 3 - (3 - \sqrt{2}) = \sqrt{2} \]

Step 4: Write the third term

Use the common difference to find the third term by adding \(d\) to the second term \(a_2\): \[ a_3 = a_2 + d = 3 + \sqrt{2} \]

Step 5: Write the fourth term

Add the common difference \(d\) to the third term to find the fourth term: \[ a_4 = a_3 + d = (3 + \sqrt{2}) + \sqrt{2} = 3 + 2\sqrt{2} \]

Step 6: Write the fifth term

Finally, add the common difference \(d\) to the fourth term to find the fifth term: \[ a_5 = a_4 + d = (3 + 2\sqrt{2}) + \sqrt{2} = 3 + 3\sqrt{2} \]

Step 7: List the first five terms of the sequence

The first five terms of the arithmetic sequence are: \[ 3 - \sqrt{2}, 3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2} \]

Key Concepts

Understanding Common Difference

In an arithmetic sequence, the common difference is a crucial factor. It tells you how much each term in the sequence increases (or decreases) from the previous one. You calculate this by subtracting the first term from the second. This difference, once identified, stays consistent throughout the entire sequence, which is what makes the sequence arithmetic.

For the given sequence, the first term is \(a_1 = 3 - \sqrt{2}\) and the second term is \(a_2 = 3\). To find the common difference \(d\), you subtract the first term from the second:

• \(d = a_2 - a_1 = 3 - (3 - \sqrt{2}) = \sqrt{2}\)

Now, you know that each step from one term to the next involves adding \(\sqrt{2}\). This step is essential because it sets the pace for reaching each subsequent term in the sequence.

Introduction to the First Term

The first term in an arithmetic sequence is the initial value from which we start our sequence. It usually is denoted as \(a_1\). In simpler terms, this is like the starting point of a journey where each step follows a consistent pattern, defined by the common difference.

In this particular example, the first term is given as \(3 - \sqrt{2}\). This term shows where our sequence begins. Understanding this starting point is crucial because it sets up the entire sequence. Without it, you can't accurately determine the subsequent terms even if the common difference is known.

Sequence Terms Calculation

Calculating the terms of an arithmetic sequence is a straightforward process once you know the first term and the common difference. Each term is derived by adding the common difference to the previous term.

Let's see how this works using the provided first term \(a_1 = 3 - \sqrt{2}\) and the common difference \(d = \sqrt{2}\):

• The second term \(a_2\) is given directly as \(3\).
• To find the third term \(a_3\), add \(\sqrt{2}\) to the second term: \(a_3 = a_2 + d = 3 + \sqrt{2}\).
• The fourth term \(a_4\) continues the pattern: \(a_4 = a_3 + d = 3 + 2\sqrt{2}\).
• Finally, the fifth term \(a_5\) is found by adding \(\sqrt{2}\) to \(a_4\): \(a_5 = 3 + 3\sqrt{2}\).

Thus, once you grasp the concept of starting from a first term and continually applying the common difference, you can easily map out any number of terms in an arithmetic sequence.