Introduction to Arithmetic Sequences

Arithmetic sequences are a fundamental concept in mathematics, often seen in real-life applications such as calculating distances between fence posts or understanding patterns in numerical series. In an arithmetic sequence, each term after the first is obtained by adding a constant value known as the common difference.

The formula for the n-th term of an arithmetic sequence is:

a_n  a_1   (n - 1)d

where:

a_n is the n-th term of the sequence. a_1 is the first term of the sequence. d is the common difference. n is the total number of terms in the sequence.

Problem Statement

The problem at hand involves determining the common difference in an arithmetic sequence where the first term is 3 and the last (7th) term is 15. There are a total of 7 terms in the sequence.

Step-by-Step Solution

Let's solve the problem using the provided values:

a_1 3 a_7 15 n 7

Substitute these values into the formula:

15  3   (7 - 1)d

Simplifying the equation:

15  3   6d12  6dd  2

The common difference, d, is 2.

Real-World Application

Imagine you are calculating the distance between fence posts. If there are 5 posts between the first and last posts, there will be 6 gaps. The total distance between the first and last post is 12 (15-3), and dividing this by the number of gaps (6) gives the common difference, which is 2. Thus, the sequence of fence post distances is:

3, 5, 7, 9, 11, 13, 15

Alternative Formula

The same result can also be obtained using the alternative formula:

d  frac{t_m - t_n}{m - n}  frac{15 - 3}{7 - 1}  2

This formula is directly derived from the slope formula, where d represents the slope with a run of 1.

Conclusion

By understanding the formula for an arithmetic sequence and utilizing given values, we can easily find the common difference. This concept is not only useful in mathematics but also in practical applications such as planning the layout of buildings, measuring distances, and even in financial calculations. Remember, the common difference, d, is the constant added to each term to get the next term in the sequence.