Introduction to Sequences

Definition of Sequences

A sequence is an ordered list of numbers arranged in a specific order. Each number in the sequence is called a term or element. The order is important because the position of each term affects its value.

Sequences can be:

• Finite sequences – these have a limited number of terms, for example, 5 terms only.
• Infinite sequences – these continue indefinitely without an end.

For example, the sequence 2, 4, 6, 8, 10 is finite with 5 terms. The sequence 1, 2, 3, 4, 5, ... goes on forever and is infinite.

Notation and Terminology

The general term (or nth term) of a sequence is a formula that allows you to find any term in the sequence without listing all the previous terms.

Terms in a sequence are often written as $u_n$ or $a_n$, where $n$ represents the position (or index) of the term in the sequence.

For example, $u_1$ is the first term, $u_2$ is the second term, and so on. The number $n$ must be a positive integer.

Knowing the position helps to identify or calculate terms directly using the general term formula.

For example, for the sequence 2, 4, 6, 8, the general term is $u_n = 2n$.

Types of Simple Sequences

Arithmetic sequences are sequences where the difference between consecutive terms is constant. This difference is called the common difference.

For example, in the sequence 3, 7, 11, 15, 19, the difference between each term is 4.

Geometric sequences are sequences where each term is found by multiplying the previous term by a constant number called the common ratio.

For example, in the sequence 2, 6, 18, 54, 162, each term is multiplied by 3 to get the next term.

Note that not all sequences are arithmetic or geometric; some follow other patterns.

Recognising whether a sequence is arithmetic or geometric helps to understand its pattern and find terms easily.

Finding Terms and Patterns

To find specific terms in a sequence, you need to identify the pattern or use the general term formula if given.

For arithmetic sequences, the general term formula is:

$$u_n = a + (n - 1)d$$

where:

• $a$ is the first term
• $d$ is the common difference
• $n$ is the term position

For geometric sequences, the general term formula is:

$$u_n = a \times r^{n - 1}$$

where:

• $a$ is the first term
• $r$ is the common ratio
• $n$ is the term position