### Patterns and Rules

A **sequence** is an ordered list of numbers. Each number in a sequence is called a **term of a sequence**. The terms in a sequence usually follow a pattern.

- In the sequence 1, 4, 7, 10, 13, …, each term is 3 less than the next term.
- In the sequence 3, 6, 12, 24, 48, …, each term is $\frac{1}{2}$ of the next term.

The sequence 2, 5, 10, 13, 26 is a **finite sequence** because it has a limited number of terms. The sequence 20, 10, 5, 2.5, 1.25, … is an **infinite sequence** because it continues without end. The ellipses (…) at the end of the sequence indicate that it goes on forever.

A **recursive rule** for a sequence tells how to find any term in the sequence by using the previous term. An **explicit rule** for a sequence tells how to find any term in the sequence directly, given its position in the sequence. The terms of a sequence are often written as $a_1, a_2, a_3,...a_n,...$ where the subscript is the position of the term in the sequence.

### Arithmetic Sequences

In an **arithmetic sequence**, there is a constant difference between consecutive terms. This means the difference between a term and the previous term is always the same value. The difference between consecutive terms in an arithmetic sequence is called the **common difference**.

Determine the common difference. Subtract each term from the next term.

0, 5, 10, 15, 20, 25, 30, …Each term is 5 greater than the previous term. So, the common difference $d$ is 5.

Use either the recursive rule or explicit rule to find the given term.

The question asks for the 21st term, which would require multiple calculations. It will be more efficient to use the explicit rule.

Substitute $n=21$.### Geometric Sequences

In a **geometric sequence**, there is a constant ratio between consecutive terms. This means that the ratio between a term and the previous term is always the same value. The ratio between consecutive terms in a geometric sequence is called the **common ratio**.

Identify the common ratio. Divide each term by the previous term:

64, 96, 144, 216, …Use either the recursive rule or explicit rule to find the given term.

The question asks for the 10th term, which would require multiple calculations. It will be more efficient to use the explicit rule.

Substitute $n=10$.The recursive rule is $a_1=64$ and $a_n=1.5 \cdot a_{n-1}$.

The explicit rule is:### Other Sequences

Sequences that are neither arithmetic nor geometric may be best described in words as repeating patterns. In the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3 …, the numbers 1, 2, 3 repeat forever. In the sequence 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, …, an additional zero is placed between each pair of ones.

The **Fibonacci sequence** is a sequence of numbers in which each term is found by adding the two previous terms. The first several terms of the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ….

The Fibonacci sequence is an example of a recursively defined sequence that does not have a common difference or common ratio. The recursive rule for the Fibonacci sequence is $a_1=1, a_2=1,$ and $a_n=a_{n-1}+a_{n-2}$.

Many things in nature grow according to this sequence. The number of leaves on a small plant or petals on a flower often come in numbers that match terms in the Fibonacci sequence.