Maths: In honour of Sophie Germain

This morning I awoke with an interesting numbers idea, which is probably not novel, but I had not come across it before. Arising and looking at my mail, Jan Pittman had sent me a link about Sophie Germain, and how her parents took away her candles at night, to stop her doing mathematics.

It struck me that I had needed no candles (and I needed no abacus, either: this was just a thought thing)

Take several consecutive cubes, like 1^3, 2^3 and 3^3. Their values are 1, 8 and 27.

Add these together, and you get 36, which is the square of (1+2+3)^2.

This relationship holds for larger number sets, but only if you start at 1.

I am sure this is a theorem that has a name, and that somebody has long since proved it, but I can not see how to prove it: it just is, but in mathematics, that will not do. Please prove it for me, but don't publish it in the comments: leave the puzzle for others. Just say that you solved it, OK? You can send me a message if you work it out.

Note: my good friend Peter Chubb told me the name of the theorem, but I am not sharing, beyond saying the discoverer may have been called Nic by his friends :-) 

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