Question

How do I find the common difference of an arithmetic sequence and provide an example?

Answer 1

To find the common difference of an arithmetic sequence, you subtract any term from the term that follows it. The common difference (often denoted as \(d\)) is constant throughout the sequence.

The formula to find the common difference is:

\[
d = a_{n} - a_{n-1}
\]

where \(a_{n}\) is the nth term and \(a_{n-1}\) is the term just before it.

### Example

Consider the arithmetic sequence: \(4, 7, 10, 13, 16\).

1. Let's identify two consecutive terms. Taking the first two terms:
- \(a_1 = 4\)
- \(a_2 = 7\)

2. Calculate the common difference:
\[
d = a_2 - a_1 = 7 - 4 = 3
\]

3. To confirm, let's check the difference between the next terms:
- \(a_3 = 10\)
\[
d = a_3 - a_2 = 10 - 7 = 3
\]
- \(a_4 = 13\)
\[
d = a_4 - a_3 = 13 - 10 = 3
\]

As you can see, the common difference in this arithmetic sequence is \(3\). Each term is formed by adding \(3\) to the previous term.