Problem 18 - Algebra for College Students

Question

$$ 4,2,0,-2,-4, \ldots \quad-2 n+6 $$

Step-by-Step Solution

Answer

The sequence follows an arithmetic pattern described by \(-2n + 6\).

Step 1: Identify the Pattern

First, observe the sequence: 4, 2, 0, -2, -4, .... Notice that each term is decreasing by 2. This means the sequence is an arithmetic sequence with a common difference of -2.

Step 2: Determine the General Formula

The general formula for the nth term of an arithmetic sequence is given by \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference. In this sequence, \( a_1 = 4 \) and \( d = -2 \), so the formula becomes \( a_n = 4 + (n-1)(-2) \).

Step 3: Simplify the Formula

Simplify the equation from Step 2:\[a_n = 4 + (n-1)(-2) = 4 - 2n + 2 = -2n + 6.\] This matches the formula given in the problem, indicating the sequence is correctly described by \( -2n + 6 \).

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is the key to understanding how the sequence grows or shrinks. It is the constant number you add or subtract to move from one term to the next in the sequence. For example, consider the sequence 4, 2, 0, -2, -4, ... . Here, you can see that each term is obtained by subtracting 2 from the previous term, which makes the common difference \( d = -2 \). This negative common difference means the sequence is decreasing.

• If the common difference is positive, the sequence increases.
• If it's zero, all the terms will be the same.

Without understanding the common difference, it would be challenging to predict how any sequence continues beyond its initial terms.

nth Term Formula

The nth term formula of an arithmetic sequence is powerful. It allows you to find any term in the sequence without listing all the previous terms. The formula is: \[ a_n = a_1 + (n-1) \times d \] where

• \( a_1 \): the first term in the sequence.
• \( n \): the term number you want to find.
• \( d \): the common difference.

Let's take the given sequence: 4, 2, 0, -2, -4, ... . Here, \( a_1 = 4 \) and \( d = -2 \). Plug these into the formula: \[ a_n = 4 + (n-1)(-2) \] Simplify it to get \[ a_n = -2n + 6 \]. This formula efficiently describes the sequence, no matter how large \( n \) becomes.

Sequence Pattern

Recognizing the pattern in any sequence is fundamental in determining whether it is arithmetic. Patterns can be visual, numerical, or based on some form of regular occurrence. With arithmetic sequences, the pattern is based on adding the same number each time.
In our sequence of 4, 2, 0, -2, -4, ..., the consistent pattern of subtracting 2 is evident. To identify whether a sequence follows a pattern:

• Observe the sequence terms and calculate the difference between them.
• Check if this difference remains constant across all consecutive terms.

Once a pattern is identified as arithmetic, the sequence is easy to analyze and predict.