Proofs
Charles Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers:
The sum of any set of consecutive odd numbers starting from 1 is a square, and the quantity that is squared is the count of odd numbers in the sum. The latter is easily seen to be a count of the form 1+2+3+4+...+n.
In the more recent mathematical literature, Stein (1971) uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also Benjamin, Quinn & Wurtz 2006); he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof". Kanim (2004) provides a purely visual proof, Benjamin & Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs.
Read more about this topic: Squared Triangular Number
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