A sequence is a discrete function with the integers as its domain e.g. f(n)=½(n²+n) for n=1,2,3,... gives the sequence of triangle numbers 1,3,6,10,15,...

A recurrence relation (sometimes called a difference equation, iterative formula or recursive definition) is a method of calculating each term in the sequence from the previous terms e.g. the triangle numbers are also defined by the relation u_n=u_n-1+n for n=2,3,4,... where u₁=1.

Recurrence relations can be used to model discrete situations that occur in finance, population growth and other areas.

First order linear recurrence relations u_n=ku_n-1+p, include the special cases of arithmetic sequences (k=1) and geometric sequences (p=0).

Second order linear recurrence relations u_n=au_n-1+bu_n-2+p and higher order linear recurrence relations are also possible.

activity page - fractals and geometric sequences

activity page - Tower of Hanoi

sum of arithmetic sequence - Java applet demonstration

infinite geometric series
Baravelle Spirals are generated by connecting the midpoints of the successive sides of a regular polygon. The sum of an infinite geometric sequence is illustrated by the triangles formed.

1994 as a sum of consecutive positive integers - Math Forum

infinite geometric series
Baravelle Spirals are generated by connecting the midpoints of the successive sides of a regular polygon. The sum of an infinite geometric sequence is illustrated by the triangles formed.

$1000000 or 1 cent, 2 cents, 4 cents, ... for a month - Math Forum

ant walking in a squared spiral - Math Forum

animation of bouncing ball - geometric sequence?

visualising an infinite geometric series - Cynthia Lanius

hare and tortoise race - Java applet
Use infinite geometric series to explain the paradox.

elimination of medicine from the body - NCTM
interactive activity illustrating the use of recursion to model the changing amount of medicine in an athlete's body