The magic of the integers – a result found in “Ascent to Orbit”

As you know I like number theory and if I find something really interesting I’ll post it up here on the NFO site.

I was flicking through Arthur C Clarke’s “Ascent to Orbit” yesterday when I came across a mathematical result I don’t think I’ve seen before, and it’s a pretty remarkable result too.  Anyone who has done “A” level maths will know that the sum of the first n integers is given by 0.5 x n x (n+1).  If you square the result to get the square of the sum of the first n integers you get 0.25 x n2 x (n+1)2 and the amazing thing is that this is exactly the same as the sum of the cubes of the first n integers or 13 + 23 + 33 + … +n3 .  In other words (1 + 2 + 3 + … + n)2 = 13 + 23 + 33 + … + n3 .  Now isn’t that amazing?

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