Talk:Cassini and Catalan identities

Warehousing this proof. Charles Matthews 11:58, 25 October 2005 (UTC)

Direct proof, by mathematical induction

For $n=m+1$ the result must be $(-1)^{m+1}$. Replacing in the equation we have

$F_{m+1-1}F_{m+1+1}-F_{m+1}^{2}=F_{m}F_{m+2}-F_{m+1}^{2}$

Rewriting the equation for an easier understanding we have that

$F_{m}F_{m+2}-F_{m+1}^{2}$	$=-F_{m+1}^{2}+F_{m}F_{m+2}$
	$=-F_{m+1}F_{m+1}+F_{m}F_{m+2}$

Recalling the formula for the Fibonacci numbers we know that

$F_{n}=F_{n-1}+F_{n-2}$

Therefore for $n=m+1$

$F_{m+1}$	$=F_{m+1-1}+F_{m+1-2}$
	$=F_{m}+F_{m-1}$

And for $n=m+2$

$F_{m+2}$	$=F_{m+2-1}+F_{m+2-2}$
	$=F_{m+1}+F_{m}$

Replacing these two known values in the equation we now have that

$-F_{m+1}F_{m+1}+F_{m}F_{m+2}$	$=-F_{m+1}(F_{m}+F_{m-1})+F_{m}(F_{m+1}+F_{m})$
	$=-F_{m+1}F_{m}-F_{m+1}F_{m-1}+F_{m}F_{m+1}+F_{m}^{2}$
	$=-F_{m}F_{m+1}-F_{m-1}F_{m+1}+F_{m}F_{m+1}+F_{m}^{2}$
	$=-F_{m-1}F_{m+1}+F_{m}F_{m+1}-F_{m}F_{m+1}+F_{m}^{2}$
	$=-F_{m-1}F_{m+1}+F_{m}^{2}$
	$=-(F_{m-1}F_{m+1}-F_{m}^{2})$
	$=-(-1)^{m}$
	$=(-1)^{1}(-1)^{m}$
	$=(-1)^{m+1}$

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Cassini and Catalan identities. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20070206051231/http://milan.milanovic.org/math/english/pi/cassini.htm to http://milan.milanovic.org/math/english/pi/cassini.htm

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 20:17, 31 July 2017 (UTC)