Small Engines (Lawn Mowers, etc.): Craftsman Riding Mower Battery Issue, battery cables, positive cable

Question
Hi,
It all started when a new battery would last about two weeks, at which point I have to charge or jump off. Now the battery cables spark and melt through the battery terminals. I made sure the cables are hooked up correctly.  It's almost like the positive and negitive current is crossing.  Help.

Answer
A dead short that bad should be easy enough to find.  Follow the red positive cable to the solenoid and disconnect it, this is the part I am guessing is the problem, it must be shorted out internally.
It is likely visibly damaged, take this part off and go to your local
mower shop or auto zone, and ask them to test it.
If that does not prove to be bad, then get back to me with your mower and engine's model numbers.