Sequences & Functions

Sequences

A sequence is an ordered list of terms.

Linear (arithmetic) sequences have a constant difference $d$. The $n$th term is $a + (n-1)d$, or more simply $dn + (\text{constant})$. For $5, 8, 11, 14, \dots$ the $n$th term is $3n + 2$.
Quadratic sequences have a constant second difference. The $n$th term has the form $an^2 + bn + c$, where $2a$ equals the second difference.

For $2, 6, 12, 20, \dots$: first differences $4, 6, 8$; second difference $2$, so $a = 1$. Subtracting $n^2$ ($1,4,9,16$) leaves $1, 2, 3, 4$, i.e. $n$. The $n$th term is $n^2 + n$.