Skill: Write the explicit formula for an arithmetic sequence

An arithmetic sequence has a constant difference between consecutive terms. Its explicit formula is

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

where:

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

Example: Write an explicit formula for the sequence

$$7,\ 11,\ 15,\ 19,\ldots$$

Step 1: Identify the first term.

The first term is

$$a_1=7$$

Step 2: Find the common difference.

Subtract consecutive terms:

$$11-7=4$$

So,

$$d=4$$

Step 3: Substitute into the formula.

$$a_n=a_1+(n-1)d$$ $$a_n=7+(n-1)(4)$$

This is an explicit formula for the sequence. It can also be simplified:

$$a_n=7+4n-4$$ $$\boxed{a_n=4n+3}$$

For example, when $n=3$,

$$a_3=4(3)+3=15$$

which matches the third term of the sequence.