Problem 13 - Precalculus Mathematics for Calculus

Question

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$5,8,11,14, \dots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 3.

Step 1: Understanding an Arithmetic Sequence

To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant. An arithmetic sequence has a common difference, denoted as \(d\), which remains the same between any two successive terms.

Step 2: Calculate the Difference Between Consecutive Terms

Find the difference between the first and second terms: \(8 - 5 = 3\). Next, find the difference between the second and third terms: \(11 - 8 = 3\). Finally, find the difference between the third and fourth terms: \(14 - 11 = 3\).

Step 3: Analyze the Differences Found

The calculated differences are consistent: 3, 3, and 3. Since the difference remains constant, the sequence is arithmetic.

Step 4: Identify the Common Difference

Now that we have confirmed the sequence is arithmetic, the common difference \(d\) is the consistent difference found in Step 2, which is \(d = 3\).

Key Concepts

Common Difference

In an arithmetic sequence, the magic lies in the common difference, often represented by the letter \(d\). This is the number that we add (or subtract) each time to move from one term to the next. To find this nifty number, subtract any term in the sequence from the one directly following it.
For example, in the sequence given:

• Subtract the first term from the second: \(8 - 5 = 3\).
• Check it again by subtracting the second term from the third: \(11 - 8 = 3\).
• Verify once more with the third and fourth terms: \(14 - 11 = 3\).

Pretty neat, right? This consistent difference of \(3\) confirms that each number in the sequence is merely three more than the one before it. Thus, \(d = 3\) is the common difference of our sequence.

Consecutive Terms

In the sequence world, consecutive terms are simply terms that follow one after the other without any gaps in between. Imagine a train: each car follows another in a perfect line.
When dealing with arithmetic sequences, these terms display a uniform pattern, based on the common difference.
Let's look at our sequence in context. We start with 5, then we go to 8, 11, and 14 consecutively.

• The difference between 5 and the next term, 8, is 3.
• Similarly, from 8 to 11, the gap remains 3.
• And from 11 to 14, once again, the difference is 3.

Each subsequent number is just a busy little train car following the predictable three-car gap to keep things orderly and arithmetic!

Sequences

Sequences are like number stories told in order, where each number in the tale is called a term. In an arithmetic sequence, the plot thickens with a twist – each number is reached by adding the common difference to the one before it.
An arithmetic sequence starts strong and grows steadily, all thanks to that consistent common difference. Whether you start from 5 and travel through 8, 11, 14, and beyond, you are always sure of heading in a straight line. Each term is a traveler, stepping precisely three paces forward with each step.
To summarize:

• An arithmetic sequence is defined when each term is a fixed step, called the common difference, from the term before it.
• It's predictable and regular, making it easier to understand and calculate the next term.
• Once the common difference is identified, you can extend the sequence as far as you'd like, knowing each step will bring you closer to infinity!

Happy sequencing!