Sequences Use Position to Organize Values

A sequence is a function whose inputs are ordered integers. The notation $a_n$ names the value at position $n$, allowing patterns to be stated without listing every term.

Explicit and recursive descriptions

An explicit rule computes a term directly. For $a_n=3+2(n-1)$, the sequence begins $3,5,7,9,\ldots$.

A recursive rule states a starting value and how to obtain the next term:

$a_1=3,\qquad a_{n+1}=a_n+2.$

The two rules describe the same arithmetic sequence.

Index origin matters

The rule $2^n$ gives $2,4,8,\ldots$ when indexing begins at $n=1$, but $1,2,4,\ldots$ when indexing begins at $n=0$. State the starting index; it is part of the definition.

Difference and ratio tests

Constant first difference indicates an arithmetic sequence. Constant ratio indicates a geometric sequence. These tests diagnose a pattern but do not prove a rule from finitely many observations—many formulas can fit the same initial terms.

Check your understanding

Find an explicit rule for $10,7,4,1,\ldots$ with first term indexed by $1$.

Show the reasoning

The common difference is $-3$, so $a_n=10-3(n-1)=13-3n$.