Reason of a continuation

In Mathematical, the reason is the value which makes it possible to pass from a term to the following in some continuations defined by recurrence.

Arithmetic continuation

See also: arithmetic Continuation

In an arithmetic continuation $\backslash \; (u\_n)\; \_\; \{\backslash ,\; N\; \backslash ,\; \backslash \; geq\; \backslash ,\; n\_0\}$ where the reason is noted R,
the relation of recurrence is the following one:

$\backslash \; forall\; N\; \backslash \; geq\; n\_0,\; \backslash \; quad\; u\_\; \{n+1\}\; =\; u\_n\; +\; R$

Geometrical continuation

See also: geometrical Continuation

In a geometrical continuation $\backslash \; (u\_n)\; \_\; \{\backslash ,\; N\; \backslash ,\; \backslash \; geq\; \backslash ,\; n\_0\}$ where the reason is noted Q,
the relation of recurrence is the following one:

$\backslash \; forall\; N\; \backslash \; geq\; n\_0,\; \backslash \; quad\; u\_\; \{n+1\}\; =\; q.u\_n$

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