hi therussequilibrium

Finally remembered that formula and another way to prove it.

You can use a similar method for any summation of the form

Let's say you have spotted that the formula you want is

Start with the next highest power of n, ie. n cubed.

Write out this expression for n=1, n=2, n=3, ...n=n

............... ............ .............. ............. .............

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Now add up each column. Most of the cube terms on the LHS cancel with most of those on the RHS. 1+1+1...+1 comes to n.

The formula for the {triangle numbers} is

so

Re-arranging

multiply by 2 to remove the fraction and factorising out the common factor of (n+1)

and so, finally

Bob