Finding the nth Term Rule for an Arithmetic Sequence: A Practical Guide

Have you ever wondered how to determine the nth term of an arithmetic sequence? In this guide, we will explore the method to find the nth term rule for the sequence 20, 18, 16, 14. We will break down each step, explain the concept of an arithmetic sequence, and provide real-world examples to ensure a thorough understanding.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. This constant difference is what defines the arithmetic sequence and allows us to predict any term in the sequence.

Identifying the Pattern

Let's consider the given sequence: 20, 18, 16, 14. We can observe that each term decreases by 2 compared to the previous term. This constant difference, denoted as d, is -2. This is a key characteristic of an arithmetic sequence.

Formulating the nth Term Rule

The formula for the nth term of an arithmetic sequence is given by:

an a1 (n - 1) cdot d

where:

an is the nth term, a1 is the first term, and d is the common difference.

For the given sequence, we identify:

a1 20 (the first term), d -2 (the common difference).

Applying the Formula

Substituting these values into the formula, we get:

an 20 (n - 1) cdot (-2)

Simplifying this expression, we have:

an 20 - 2(n - 1)

Further simplification yields:

an 20 - 2n 2 22 - 2n

Therefore, the nth term rule for the sequence is:

an 22 - 2n

Verification

Let's verify this rule by calculating the first few terms of the sequence:

For the 1st term, n 1: a1 22 - 2(1) 22 - 2 20 For the 2nd term, n 2: a2 22 - 2(2) 22 - 4 18 For the 3rd term, n 3: a3 22 - 2(3) 22 - 6 16 For the 4th term, n 4: a4 22 - 2(4) 22 - 8 14

Understanding the Components

The first term, 20, is crucial as it establishes the starting point of the sequence. By subtracting 2n from 22, we ensure that we consistently increase the value of n, reflecting the progression of the sequence.

The common difference, -2, is the key to understanding how each term relates to the previous one. By subtracting 2 each time, we maintain the arithmetic nature of the sequence.

Conclusion

Understanding and applying the nth term rule for an arithmetic sequence is a fundamental skill in mathematics. This method allows us to predict and calculate the value of any term in the sequence, providing a powerful tool for problem-solving.

By following these steps, you can easily find the nth term rule for any arithmetic sequence. With practice, you will be able to identify patterns and solve problems more efficiently.