1. ## 100 Square problem

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?

2. Hello Cornopolis

Welcome to Math Help Forum!
Originally Posted by 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?

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

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

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