Sequences

What:

An infinitely ordered list. The items in it are called terms. EG:

• Sequence of natural numbers: $1,2,3,4,5,...$
• Scores I got when infinitely rolling die: $1,4,2,5,2,...$
• A sum of the subset of terms is called Series

Describing a pattern as a formula:

• In the following:
  • $a$ is the first term
  • $d$ is the common difference
  • $r$ is common ratio (i.e. $T_n/T_{n-1}$)
• For arithmetic sequences (constant first difference):
  • $T_n=a+(n-1)d$
  • $S_n=\frac{n}{2}[2a+(n-1)d]$
• For geometric sequences (constant second difference):
  • $T_n=a{r}^{n-1}$
  • $S_n=\frac{a(1-r^n)}{1-r}$
  • $S_{\infty }=\frac{a}{1-r}$ (Given $|r|<1$)
    “

Different Types of Sequences

Increasing, Decreasing, and Monotone Sequences

A sequence $a_n$ is increasing for all $n\ge n_0$ if

$$a_n\le a_{n+1}\text{ for all }n\ge n_0.$$

I.E. if a term is followed by a bigger term every time.

A sequence $a_n$ is decreasing for all $n\ge n_0$ if

$$a_n\ge a_{n+1}\text{ for all }n\ge n_0.$$

I.E. every term is followed by a lesser term.

A sequence $a_n$ is a monotone sequence for all $n\ge n_0$ if it is increasing for all $n\ge n_0$ or decreasing for all $n\ge n_0$.

I.E. After a certain point, the monotone sequence doesn’t change direction.