Introduction to Arithmetic Sequences

Arithmetic sequences are a fundamental part of mathematics, often appearing in various real-world applications such as finance, science, and technology. This article aims to explain how to find the nth term of a given arithmetic sequence, using the example sequence: 3, 10, 17. Understanding the concept and its application can help students and educators alike to better grasp the significance of arithmetic sequences.

Understanding the nth Term of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the preceding term. The nth term (tn) can be determined using a specific formula.

Example Problem: Finding the nth Term of 3, 10, 17

Consider the sequence: 3, 10, 17. We are tasking with finding the nth term of this sequence.

Determining the Common Difference

The first step in finding the nth term of any arithmetic sequence is to determine the common difference (d). This is done by subtracting any term from the term that follows it. In our example sequence, we have:

10 - 3 7

17 - 10 7

Since the difference between consecutive terms is constant, we confirm that this is an arithmetic sequence with a common difference (d) of 7.

Using the Formula to Find the nth Term

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

cn a1 (n - 1)d

Where:

cn a1 is the first term of the sequence, d is the common difference, and n is the term number in the sequence.

Substituting the values from our example into the formula:

tn 3 (n - 1)7

Simplifying the formula, we get:

tn 3 7n - 7

Therefore, the nth term (tn) for the sequence 3, 10, 17 is:

tn 7n - 4

Verification

Let's verify our solution with the given values:

For n 1: tn 7(1) - 4 3 For n 2: tn 7(2) - 4 10 For n 3: tn 7(3) - 4 17

The calculated values match the given sequence, confirming our solution is correct.

Conclusion

Using the nth term formula, we can efficiently find the general term of any arithmetic sequence. This method is particularly useful for predicting any term in the sequence without listing all the preceding terms. Whether you are a teacher, student, or just curious about math, understanding the nth term can greatly simplify many math-related problems.

Recommended Resources

To further enhance your understanding and practice with arithmetic sequences, consider exploring these resources:

Math Is Fun: Arithmetic Sequences Math Centre: Arithmetic Sequences Khan Academy: Arithmetic Sequences

By studying these materials, you can gain a deeper understanding of arithmetic sequences and their applications.

Now that you know how to find the nth term of an arithmetic sequence, you can apply this knowledge to many other problems and scenarios. Happy solving!