## 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.

##### More Answers:

Largest Prime FactorLargest Palindrome Product

Smallest Multiple

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