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Expert: Paul Klarreich - 3/4/2007


Question
it can be proven that 1^2 + 2^2 + 3^3 + ..... N^2 = [(n)(n+1)(2n+1)] / 6

can u please tell me how the formula above could have been acheived?

Answer
Hi, Hoyong,

Proving it by induction is straightforward, but I think you are asking how we derived it in the first place.

There is an ancient derivation of this formula due to Archimedes, which this text box is too small to contain.  You can find it at:

http://www.macalester.edu/~bressoud/pub/CBN2.pdf  

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Paul Klarreich

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I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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