Skill: Write the explicit formula for a geometric sequence

A geometric sequence is a sequence in which each term is found by multiplying the previous term by the same constant number. This constant is called the common ratio, $r$.

The explicit formula is

$$a_n=a_1r^{n-1},$$

where:

• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

Worked example

Write an explicit formula for the sequence

$$5,\ -10,\ 20,\ -40,\ldots$$

Step 1: Identify the first term.

The first term is

$$a_1=5.$$

Step 2: Find the common ratio.

Divide a term by the term before it:

$$r=\frac{-10}{5}=-2.$$

Check the next pair:

$$\frac{20}{-10}=-2,$$

so the common ratio is constant.

Step 3: Substitute into the formula.

Using $a_n=a_1r^{n-1}$:

$$a_n=5(-2)^{n-1}.$$

Therefore, the explicit formula is

$$\boxed{a_n=5(-2)^{n-1}}.$$

Step 4: Check the formula.

For $n=1$:

$$a_1=5(-2)^0=5.$$

For $n=3$:

$$a_3=5(-2)^2=5(4)=20.$$

These match the sequence.

The exponent $n-1$ shows how many times the common ratio is multiplied. A negative ratio causes the signs to alternate. In contrast, an arithmetic sequence uses repeated addition or subtraction, not repeated multiplication.