#6 Sum square difference

The sum of the squares of the first ten natural numbers is,

\[1^2 + 2^2 + ... + 10^2 = 385\]

The square of the sum of the first ten natural numbers is,

\[(1 + 2 + ... + 10)^2 = 55^2 = 3025\]

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.

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The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos.

\[\left( \sum_{i}x_i \right)^2=\sum_{i}{x_i}^2 + 2\sum_{i<j}x_i x_j\]

\[\left ( \sum_{i} x_i \right )^2 - \sum_{i} {x_i}^2 = 2 \sum_{i<j} x_i x_j\]

n = 100
summation = 0
for j in range(1, n + 1):
    for i in range(1, j):
        summation += 2 * i * j

The answer is 25164150.