To warm up, let’s start with . We want to prove that the numbers within the large square that’s from the left ad from the top sum to

First look at the large square that’s from the left and lies in the first row of large squares. As we already noted above, the two numbers in the top row of this white square are and

By some very similar reasoning we can work out that the top two numbers in the large square that’s from the left and from the top are

and

Similarly, the bottom two numbers in the white square from the left in the row of white squares are

and

Adding these four numbers gives

which we can rewrite as the product

which is exactly what we wanted.

We now move on to prove the result for any positive integer . We want to show that the sum of numbers in the square that’s from the left and from the top is equal to

Let’s call the square in question

Using similar reasoning as above, you can see that the numbers in the first row of are

times the numbers in the first row of . This means that the sum of the first row of numbers in is

Similarly, the sum of the second row of numbers in is

We can continue in this vein until we come to the sum of the last row of numbers in which is

Adding up these sums of the rows of gives

(1) |

Similarly to what we did above, we can re-write this expression as

Using the formula for the sum of the first integers, this becomes

which is equal to

And since

the sum of the numbers in is equal to

This is what we wanted to show.