Sequences can be:

• increasing
• decreasing
• periodic

Sequences can be represented by:

• the \(n^{th}\) term
• recurrence relation in the form \(x_{n+1}=f(x_n)\)

Increasing sequences

Terms in these sequences always get bigger.

For example: \(u_n=\dfrac{1}{3n+1}\) as \(u_{n+1} < u_n\) for all integer \(n\).

Decreasing sequences

Terms in these sequences always get smaller.

For example: \(u_n=2^n\) as \(u_{n+1} > u_n\) for all integer \(n\).

Periodic sequences

Terms in these sequences repeat in a cycle. The order of a periodic sequence is the number of terms in a cycle.

For example: \(u_n=\dfrac{1}{u_n}\) for \(n>1\) and \(n_1=3\) has an order of \(2\).