Problem 23 - Algebra for College Students

Question

$$ -3,-6,-9,-12,-15, \ldots $$ \(-3 n\)

Step-by-Step Solution

Answer

The rule for the sequence is \(-3n\), where n is a positive integer.

Step 1: Identify the Pattern

Look at the sequence: -3, -6, -9, -12, -15, ... and identify the pattern. Notice that each term in the sequence is the previous term subtracted by 3.

Step 2: Determine the General Rule

Realize that the sequence is essentially multiplying -3 by consecutive positive integers (1, 2, 3, 4, 5,...). Therefore, the nth term can be expressed as \(-3n\).

Step 3: Verify with Examples

Substitute n = 1, 2, 3, etc., into \(-3n\). For \(n=1\), \(-3 \times 1 = -3\); for \(n=2\), \(-3 \times 2 = -6\); and so on, confirming the pattern continues.

Key Concepts

Arithmetic Sequences

An arithmetic sequence is a type of sequence where each term after the first is generated by adding a constant difference to the previous term. This difference is known as the common difference. In the given sequence of \(-3, -6, -9, -12, -15,\ldots\), you can observe that each number is achieved by subtracting 3 from the previous term.

• The first term is \(-3\).
• The second term is \(-6\) which can be calculated as \(-3 - 3\).
• The third term is \(-9\), calculated as \(-6 - 3\), and so forth.

Understanding arithmetic sequences is crucial because they form the foundation for recognizing patterns in various fields of math. This type of sequence can be used to describe phenomena that change at consistent rates, such as monthly savings plans or increasing distances covered at a fixed pace.

Pattern Recognition

Pattern recognition plays a vital role in understanding and forming sequences. In mathematical sequences, particularly in arithmetic ones, identifying the pattern helps to quickly determine successive terms without having to perform repeated calculations. In the sequence provided, you'll notice:

• Every term is created by subtracting 3 from the previous term.
• The difference between each consecutive term remains the same throughout the entire sequence.
• This observation of consistency is key to recognizing that the sequence is arithmetic.

Recognizing such patterns allows you to predict further terms and understand the underlying structure of the sequence. This skill is not just essential in math but is widely applicable in coding, problem-solving, and logical reasoning scenarios.

General Term Formula

The general term formula in a sequence is a mathematical expression that allows you to find any term in the sequence without listing all previous terms. Understanding this concept is incredibly useful for efficiently solving sequence-related problems.For the arithmetic sequence \(-3, -6, -9, -12, -15, \dots\), the pattern suggests that every term is \(-3\) multiplied by \(n\), where \(n\) is the position of the term in the sequence. Therefore, the general formula is \(-3n\).

• For \(n=1\), the term is \(-3 \times 1 = -3\).
• For \(n=2\), the term becomes \(-3 \times 2 = -6\).
• This pattern continues indefinitely.

The value of this formula is its efficiency and accuracy. Once derived, it allows for quick computation and verification of any term in the sequence, empowering students to tackle increasingly complex problems with confidence.