My iMac won't support the latest version of iTunes which I need in order to add apps to my new iPhone5

I recently purchased an iPhone5.  When I tried to sync it using my iMac with Lion, I was told to upgrade my iTunes.  I have the latest version of iTunes which my iMac will support but that won't work to sync my iPhone.  Are there other ways to sync my iphone? 
Thanks for any ideas/hellp.

You're Tag says you are running Leopard 10.5.8
If this is correct... Read on...
The first step in Upgrading... is to Snow Leopard = OS X 10.6.x
It is Not available as a download... It is a Paid Upgrade.
Do this first...
Check that your Mac meets the System Requirements for Snow Leopard...
Snow Leopard Tech Specs
http://support.apple.com/kb/SP575
If so... Purchase a Snow Leopard Install Disc...
http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard
Other countries...
http://support.apple.com/kb/HE57
After the Successful Install, run Software Update to get the latest updates for Snow Leopard.
Be sure to make a Backup of your Current System Before Upgrading...

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