# Any large square for general k

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  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.