Skill: Determine terms of an arithmetic sequence

An arithmetic sequence changes by the same amount each time. This fixed amount is called the common difference, $d$.

To find a particular term, use

$$a_n=a_1+(n-1)d$$

where:

• $a_n$ is the term you want,
• $a_1$ is the first term,
• $n$ is the term number,
• $d$ is the common difference.

Example

Find the sixth term of the sequence

$$18,\ 15,\ 12,\ 9,\ldots$$

Step 1: Identify the first term and common difference.

The first term is

$$a_1=18$$

Each term decreases by $3$, so

$$d=-3$$

Step 2: Substitute $n=6$ into the formula.

$$a_6=a_1+(6-1)d$$ $$a_6=18+5(-3)$$

Step 3: Simplify.

$$a_6=18-15=3$$

Therefore, the sixth term is

$$\boxed{3}$$

The expression $(n-1)$ is used because the first term is already given; to reach the sixth term, the common difference is added five times.