Hello Cornopolis

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**Cornopolis** Hi, I'm wondering if someone can help with a problem I have.

If you have a hundred square, 10x10 with numbers 1 - 10 in the first row etc., and you have 2x2, 3x3 squares etc. starting from 1 (top left), is there an equatuion for working out the sum total of the numbers within the square, so I could work out the total of the numbers within an nxn square?

Thanks in advance.

On row 1, the sum is:$\displaystyle 1+2+..+n=\tfrac12n(n+1)$

On row 2:$\displaystyle (10+1)+(10+2)+...+(10+n) = 10n + \tfrac12n(n+1)$

On row 3:$\displaystyle (2\times10+1)+(2\times10+2)+...+(2\times10+n) = 2\times10n + \tfrac12n(n+1)$

...

On row $\displaystyle n$: $\displaystyle ([n-1]\times10+1)+([n-1]\times10+2)+...+([n-1]\times10+n) = [n-1]\times10n + \tfrac12n(n+1)$

So the total is:$\displaystyle (0+1+ 2 + [n-1])10n+n\times\tfrac12n(n+1)$$\displaystyle =\tfrac12(n-1)n\times10n +\tfrac12n^2(n+1)$

$\displaystyle =\tfrac12n^2(10n-10+n+1)$

$\displaystyle =\tfrac12n^2(11n-9)$

Grandad