Skill: Determine the common difference of an arithmetic sequence

In an arithmetic sequence, the same amount is added or subtracted each time. This amount is called the common difference, usually written as $d$.

If two consecutive terms are given, subtract:

$$d=\text{next term}-\text{previous term}$$

If the terms are not consecutive, divide the change in the terms by the change in their positions:

$$d=\frac{\text{change in term values}}{\text{change in positions}}$$

Example

Suppose an arithmetic sequence has

$$a_3=11 \quad \text{and} \quad a_8=31.$$

Find the common difference.

Step 1: Find the change in the term values.

$$31-11=20$$

Step 2: Find the change in the positions.

From the third term to the eighth term:

$$8-3=5$$

There are $5$ equal steps between the terms.

Step 3: Divide.

$$d=\frac{20}{5}=4$$

Therefore, the common difference is

$$\boxed{4}$$

This means that $4$ is added to each term to get the next term. The common difference is an additive change, unlike the common ratio in a geometric sequence.