Sequences and Notation

Sequences as indexed rules

Sequences are defined by a rule that assigns a term to each index $n$. A sequence is a function from positive integers (often starting at $1$) to the values of the sequence. The starting index can be $0$ or any integer, but many definitions begin at $1$; shifting the start may require redefining the rule.

Notation and indexing

Let $a<em>1, a</em>2, a<em>3, \dots$ be the terms. The nth term is denoted by $a</em>n$. If the sequence is given by a function $f(n)$, then $a<em>1 = f(1)$ and generally $a</em>n = f(n)$. The index tells you the position in the ordered list. For example, $a_{17} = f(17)$.

Example: a rational sequence

If $T<em>n = \frac{n}{n+1}$, then the terms are rational numbers obtained by evaluating the rule at each $n$. So the nth term is $T</em>n = \frac{n}{n+1}$.

Special type: Arithmetic sequences

An arithmetic sequence is essentially linear: the difference between successive terms is constant. A common explicit form is $a<em>n = a</em>1 + (n-1)d$, where $d$ is the common difference.

Key takeaway on indexing

Starting index is a convention; the concept is the rule $a_n = f(n)$ applied to the index $n$. You may adjust the domain or the function if you change the starting point.