# 100 Square problem

• Dec 9th 2009, 10:28 AM
Cornopolis
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?

• Dec 10th 2009, 03:14 AM
Hello Cornopolis

Welcome to Math Help Forum!
Quote:

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

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

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