Problem 4 - Contemporary Precalculus

Question

Determine whether the sequence is arithmetic or not. If it is, find the common difference. $$-9,-6,-3,0, \dots$$

Step-by-Step Solution

Answer

Sequence: $$-9, -6, -3, 0, \dots$$ Answer: Yes, the given sequence is arithmetic with a common difference of 3.

Step 1: Identify the sequence terms

The sequence given is $$-9,-6,-3,0, \dots$$ The terms (numbers in the sequence) are -9, -6, -3, and 0.

Step 2: Check if the sequence is arithmetic

To check if the sequence is arithmetic, find the differences between consecutive terms in the sequence and see if they are constant. If they are, the sequence is arithmetic.

Step 3: Calculate the differences between consecutive terms

Calculate the differences between consecutive terms as follows: Difference between the first and second term: $$-6 - (-9) = -6 + 9 = 3$$ Difference between the second and third term: $$-3 - (-6) = -3 + 6 = 3$$ Difference between the third and fourth term: $$0 - (-3) = 0 + 3 = 3$$

Step 4: Determine if the sequence is arithmetic

As we can see, the differences between consecutive terms are all equal to 3. Since the differences are constant, the sequence is arithmetic.

Step 5: Identify the common difference

The common difference between consecutive terms in an arithmetic sequence is the constant difference we found in Step 4. In this case, the common difference is 3.

Key Concepts

Common Difference

In the world of arithmetic sequences, one of the key components is the "common difference." This concept is all about understanding the difference between one term and the next in a sequence of numbers.

To determine whether a sequence is arithmetic, you need to check if this difference remains the same as you move from one term to the next.

• For instance, look at the sequence: \[-9, -6, -3, 0, \ldots\] Calculating the difference between each pair of consecutive terms can reveal a lot.
• From \(-9\) to \(-6\), the difference is \(-6 - (-9) = 3\).
• From \(-6\) to \(-3\), the difference remains the same: \(-3 - (-6) = 3\).
• Finally, from \(-3\) to \(0\), we again find \(0 - (-3) = 3\).

With a constant difference of 3 among all consecutive terms, the sequence is confirmed as arithmetic. The common difference in this case is 3, setting the pattern for the entire sequence.

Sequence Analysis

Sequence analysis involves looking closely at a sequence of numbers to understand its properties. When analyzing sequences, particularly arithmetic ones, our goal is to determine what makes them tick.

• First, identify the terms in the sequence. In our example, these are \(-9, -6, -3, 0\).
• Next, calculate the difference between each consecutive term. This will help you see if there is a consistent pattern — a key step in identifying an arithmetic sequence.
• Check if all the differences are the same. If they are equal, the sequence is arithmetic.

Through this systematic analysis, you can decide whether a given sequence is arithmetic or not. If it is arithmetic, then you'll also identify a constant difference, which is crucial in describing how the sequence progresses.

Precalculus Concepts

Precalculus is the stepping stone to calculus, and understanding sequences is part of this journey. Arithmetic sequences form a basic yet fundamental concept.

• Such sequences exhibit evenly spaced terms, determined by the common difference.
• Learning about arithmetic sequences helps in grasping more advanced topics later in calculus.
• Concepts like summation and series are closely related and naturally follow from a solid understanding of sequences.

In an arithmetic sequence, each term is crafted by adding the common difference to the previous term. This principle is not only vital in mathematics but also sometimes used in computer science and economics. Similarly, understanding these precalculus concepts lays a strong foundation for tackling calculus, where sequences often feature in series and limits.