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.

We can solve this problem using the formulae for the sum of squares and sum of a sequence of natural numbers.

1. The sum of squares of the first n natural numbers is given by the formula:

$$n(n + 1)(2n + 1) / 6$$

Applying this formula for the first 100 numbers gives:

$$100*101*201 / 6 = 338350.$$

2. The sum of the first n natural numbers is given by the formula:

$$n(n + 1) / 2$$

The square of this sum for the first 100 numbers is:

$$(100*101 / 2)^2 = 25502500.$$

The difference between the square of the sum and the sum of the squares is:

$$25502500 – 338350 = 25164150.$$

So the difference between the sum of the squares of the first hundred natural numbers and the square of the sum is 25164150.

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