Unveiling the nth Term Rule of a Linear Sequence: A Comprehensive Guide

Understanding the nth term rule of a linear sequence can be a vital skill for students and professionals alike, particularly in fields involving mathematics and data analysis. This guide aims to simplify the process of identifying the nth term rule for a specific sequence and provide insights into related concepts.

Identifying the Sequence

Consider the linear sequence: 22, 18, 14, 10, 6. Our goal is to find the nth term rule for this sequence. The first step in this process is to identify the pattern or rule governing the sequence.

Step 1: Identify the First Differences

Now, let’s look at the first differences between consecutive terms:

18 - 22 -4 14 - 18 -4 10 - 14 -4 6 - 10 -4

The first differences are constant at -4, indicating that this sequence is linear.

Step 2: Determine the Nth Term Formula

The general form of a linear sequence is given by:

an a1 (n - 1) d

Where:

a1 is the first term d is the common difference n is the term number

For the given sequence, we can identify:

a1 22 -4

Substituting these into the general formula:

an 22 (n - 1) (-4)

Expanding this:

an 22 - 4 n 4

Simplifying further:

an 26 - 4 n

Step 3: Solution Verification

To verify, we can use the nth term rule to find subsequent terms:

26 - 4n

For n 2:

26 - 4(2) 18

For n 3:

26 - 4(3) 14

For n 4:

26 - 4(4) 10

For n 5:

26 - 4(5) 6

And so on... This matches the given sequence.

Generalizing the nth Term Rule

The nth term rule for a linear sequence is always of the form an a1 (n - 1) d, where a1 and d can be positive or negative whole or fractional. To find the specific values of a1 and d, you need to:

Find the common difference, d, by subtracting consecutive terms.

Use a1 as the initial term of the sequence.

Multiply d by n - 1 and subtract or add from a1 to get the general form of the nth term rule.

Conclusion

Understanding and applying the nth term rule is essential for analyzing and predicting sequences in various applications. Whether you’re working on homework or real-world problems, being able to find the nth term rule for a linear sequence can significantly enhance your problem-solving skills. Remember that the key steps involve identifying the common difference, using the first term, and formulating the general rule. Practice with different sequences to ensure you grasp the concept thoroughly.