proof of Cassini’s identity
For all positive integers , let denote the
Fibonacci number, with . We will show by
induction
that the identity
holds for all positive integers .
When , we can substitute in the values for ,
and yielding the statement , which is true.
Now suppose that the theorem is true when ,
for some integer .
Recalling the recurrence relation for the Fibonacci numbers,
, we have
by the induction hypothesis. So we get , and the result is thus true for . The theorem now follows by induction.
Title | proof of Cassini’s identity |
---|---|
Canonical name | ProofOfCassinisIdentity |
Date of creation | 2013-03-22 14:44:40 |
Last modified on | 2013-03-22 14:44:40 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 24 |
Author | yark (2760) |
Entry type | Proof |
Classification | msc 11B39 |
Related topic | CatalansIdentity |