Problem 18 - Contemporary Precalculus

Question

Show that the sequence is arithmetic and find its common difference. $$\\{1.5+1.5 n\\}$$

Step-by-Step Solution

Answer

If yes, what is the common difference? Answer: Yes, the given sequence is arithmetic with a common difference of 1.5.

Step 1: Identify the given sequence

The given sequence is: $$\\{1.5+1.5 n\\}\,,$$ where $$n$$ is a non-negative integer.

Step 2: Find consecutive terms of the sequence

Let's find the first few terms of the sequence by substituting n with 0, 1, 2, and 3: $$\\ a_{0} = 1.5 + 1.5 (0) = 1.5\\$$ $$\\ a_{1} = 1.5 + 1.5 (1) = 3\\$$ $$\\ a_{2} = 1.5 + 1.5 (2) = 4.5\\$$ $$\\ a_{3} = 1.5 + 1.5 (3) = 6\\$$

Step 3: Calculate the difference between consecutive terms

Now we'll calculate the difference between the consecutive terms to see if it's constant. $$\\ a_{1} - a_{0} = 3 - 1.5 = 1.5\\$$ $$\\ a_{2} - a_{1} = 4.5 - 3 = 1.5\\$$ $$\\ a_{3} - a_{2} = 6 - 4.5 = 1.5\\$$

Step 4: Verify if the common difference is constant

From Step 3, we can see that the difference between consecutive terms is constant and equal to 1.5. Therefore, the given sequence is arithmetic with a common difference of 1.5.

Key Concepts

Common Difference

In an **arithmetic sequence**, the common difference is the amount added or subtracted to each term to get to the next one. This is a key feature, distinguishing arithmetic sequences from other types.

To find the common difference of a sequence, you simply take any two consecutive terms and subtract the first from the second. For the sequence given in the exercise, we have:

• First term (\(a_0\)) is 1.5.
• Second term (\(a_1\)) is 3.
• The difference between them is \(3 - 1.5 = 1.5\).

This calculation shows that the common difference is 1.5. This consistency in the subtraction between any two consecutive terms confirms that the sequence is indeed arithmetic.

Recognizing this common difference helps us predict future terms or find specific terms without listing them all, which can be very handy, especially with large sequences.

Consecutive Terms

Consecutive terms are the elements of a sequence that come one after another. Understanding the connection between consecutive terms is crucial when dealing with sequences, especially arithmetic ones.

In the given sequence \(\{1.5, 3, 4.5, 6, \ldots\}\), each term is formed by adding the common difference to the previous term. For example:

• The first term is 1.5.
• Adding the common difference (1.5) gives us the second term: 1.5 + 1.5 = 3.
• Continue adding 1.5 for subsequent terms: 3 + 1.5 = 4.5, 4.5 + 1.5 = 6, etc.

This pattern allows us to generate each term systematically. It's like building blocks, where each piece is placed on top of the other with equal spacing (the common difference).

Knowing how to find consecutive terms gives you insight into how the sequence progresses and allows you to extend the sequence as far as needed.

Sequences and Series

Sequences are ordered lists of numbers following a specific pattern. An **arithmetic sequence**, like in our exercise, is characterized by a consistent common difference between terms.

A series, meanwhile, is the sum of the terms of a sequence. If we were to add all the terms of our arithmetic sequence up to a certain point, it becomes an arithmetic series. This can be useful in various calculations and has its own formula for quick computation. By understanding the nature and structure of sequences, we get a better grasp on how they behave and their applications.

• **Sequence:** \(1.5, 3, 4.5, \, \ldots\)
• **Series:** Sum of the terms, like \(1.5 + 3 + 4.5 + \, \ldots\)

In both cases, recognizing a common pattern is key. For anyone working with numbers routinely, like scientists or economists, sequences and series form fundamental tools for analyzing trends and behaviors.