First of all, please take a close look at the image above and analyze what's going on out there. I had discovered it accidentally in the year 2005, while playing with numbers (when I was in 3rd Grade). As you can clearly see in the image, there is a definite pattern to what happens in the successive difference series. I want a proof for my conjecture that for \(n\)th powers, the constant terms obtained would be \(n!\). Please post the complete solution, if you have proved it.