*Recursive* vs. *inductive* definition I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition:
Suppose you have sequence of real numbers. Define $a_0:=2$ and $a_{i+1}:=\frac{a_i a_{i-1}}{5}$. (Of course this is just an example and as such has only illustrative character - I could have as well taken as an example a definition of some family of sets)
I claimed that this definition was recursive since we have an $a_{i+1}$ and define it going "downwards" and use the word "inductive" only as an adjectiv for the word "proof", but my professor insisted that we distinguish between these types of definition and that this was an inductive definition, since we start with $a_0$ and work "upwards".
Now, is there even someone who can be right  ? Since to me it seems that mathematically every recursive definition is also inductive (whatever these two expressions may finally mean), since the mathematical methods used to define them (namely equations) are the same. (Wikipedia also seems to think they are the same - but I trust a sound mathematical community, i.e. you guys, more than Wikipedia)
And if there is a difference, who is right and what is, if the above is a recursive definition, an inductive definition (and vice-versa) ?
(Please, don't ask me to ask my professor again - or anything similar, since I often get this answer here, after mentioning that this question resulted from a discussion with some faculty member - since out discussion ended with him saying that "it definitely is inductive, but I just can't explain it") 

Here is my inductive definition
of the cardinality
of a finite set
(since,
in my mind,
finite sets are built
by adding elements
starting with the
empty set):
$|\emptyset|
= 0
$.
$|A \cup {x}|
=
\begin{cases}
x \in A
&\implies |A|\\
x \not\in A
&\implies |A|+1\\
\end{cases}
$
