A sequence is a list of numbers in order, like $5, 8, 11, 14, \ldots$ Each number is a term, and its place in line is its position. Seen the right way, a sequence is just a function.
Por qué funciona
Pair each position with its term: position $1$ gives $5$, position $2$ gives $8$, position $3$ gives $11$. Each input (a position) has exactly one output (a term), so this is a function whose domain is the counting numbers $1, 2, 3, \ldots$ We write $a_n$ instead of $a(n)$, but it means the same: $a_4 = 14$ is the output when the input is $4$. Its graph is a row of separate dots, one above each whole number, not a connected curve.
There are two ways to describe a sequence. An explicit rule gives any term from its position, such as $a_n = 3n + 2$. A recursive rule gives the first term and how to get each term from the one before, such as $a_1 = 5$ and $a_n = a_{n-1} + 3$.
Idea clave
A sequence is a function on the positions $1, 2, 3, \ldots$ An explicit rule turns a position into a term; a recursive rule builds each term from the previous one.
Worked examples
Ejemplo resuelto: An explicit rule
For $a_n = 3n + 2$, find $a_{10}$, and find which term equals $47$.
- $a_{10} = 3(10) + 2 = 32$.
- Set $3n + 2 = 47$: $3n = 45$, so $n = 15$. The $15$th term is $47$.
Respuesta: $a_{10} = 32$; $a_{15} = 47$.
Ejemplo resuelto: A recursive rule
A sequence has $a_1 = 4$ and $a_n = 2a_{n-1} - 1$. Find $a_4$.
- $a_2 = 2(4) - 1 = 7$.
- $a_3 = 2(7) - 1 = 13$.
- $a_4 = 2(13) - 1 = 25$.
Respuesta: $a_4 = 25$
Mind the difference between a position and a term: in $5, 8, 11, \ldots$ the third term is $11$, not $3$.