The sum of the squares of the first ten natural numbers is,
The square of the sum of the first ten natural numbers is,
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
There is a way to get the result with just some calculations -- and I did not find it until after solving this problem with Excel. I already knew that the sum of the numbers 1,2, ..., n is n(n+1)/2 but what I did not know is that the sum of 12,22,...,n2 is n(n+1)(2n+1)/6 -- see http://en.wikipedia.org/wiki/Square_pyramidal_number
This is also easy to do in Excel. In 100 contiguous cells in some column (say A1:A100) enter the numbers 1, 2, ..., 100. There are several ways to do this. Enter 1 and 2 in A1 and A2 respectively. Then, select A1:A2, click the right-bottom corner of the cell and drag down to A100. Or, enter 1 in A1 and =A1+1 in A2. Select A2. Next, open the GoTo dialog box (short cut is CTRL+g), enter A100 as the destination and hold SHIFT while clicking OK. Fill down the contents of A2 with CTRL+d There are still other ways to get this series in Excel.
To get the sum of the squares, in B1 enter =A1^2. If Excel doesn't automatically fill B2:B100 with the formula do so yourself. In B101 enter the formula to sum the numbers above.
To get the square of the sums, sum the values in A1:A100 in A101 and in A102 enter the formula to square the result in A101.
