Diagonals of a square array

The 5 \times 5 array

contains 5^2 = 25 dots, arranged in 5 rows and 5 columns.

We can view the array as consisting of a series of disjoint diagonals:

Counting the dots in the diagonals we get

(1+2+3+4)+5+(4+3+2+1) = 5\times 5

which, looked at another way, says

1+2+3+4 = (5^2-5)/2 .

Of course, there’s nothing special about 5: we can see by looking at an 11\times 11 square array of dots that

1+2+\ldots + 9 +10 = (11^2-11)/2

and, more generally, if n is any positive integer then

1+2+\ldots + n-2 +n-1 = (n^2-n)/2 .

Seeing a square array of dots as a series of diagonals is key to understanding, and proving, that

1+2+\ldots + n-2 +n-1 = (n^2-n)/2

for any positive integer n .

This idea is a small, yet significant, step in awakening a developing mathematical brain.


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