Main Index Number Theory Sequences Recurrent sequences Linear recurrent sequences First order sequences
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First order recurrence relations

The first order recurrence relation (sequence) has form

x _ (n + 1) = f(x _ n, n) , with      x _ 0 = a _ 0 .(1)

An autonomous first order recurrence

x _ (n + 1) = f(x _ n) , with      x _ 0 = a _ 0 .(2)

is actually a map. If it is in addition linear, then it may have the form

x _ n = r x _ (n - 1), n > 0, with x _ 0 = A .

Its general solution is typeset structure, which is a geometric sequence with ratio typeset structure.

One possibility for the non-homogeneous case is

x _ n = r x _ (n - 1) + c, n > 0, with x _ 0 = A .

Here typeset structure is a constant. The solution depends on typeset structure. If typeset structure it can be written in the form

x _ n = A r^n + c Underoverscript[∑, i = 1, arg3] r^(n - i) = A r^n + c (r^n - 1)/(r - 1) .

If typeset structure then typeset structure

Cite this web-page as:

Štefan Porubský: First order sequences.


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