@angeliccanary Yep, that's exactly how you would rigorously prove that a pattern exists!

Just a really small note- for the proof, it actually doesn't strictly matter that n(n-1) is the LCD; we can still algebraically set n(n-1) to be the common denominator no matter what. This is sort of like how we can write 1/2 + 3/5 as 22/20, even though technically it's 11/10 simplified. You're right that n(n-1) is the LCD because the numerator becomes 1, though

Great job!