eulerbooks/problem0006.ipynb

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    "# Sum Square Difference\n",
    "> The sum of the squares of the first ten natural numbers is,\n",
    "> $$1^2 + 2^2 + \\cdots + 10^2 = 385$$\n",
    "> The square of the sum of the first ten natural numbers is,\n",
    "> $$(1 + 2 + \\cdots + 10)^2 = 55^2 = 3025$$\n",
    "> Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025 - 385 = 2640$.\n",
    "> \n",
    "> Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.\n",
    "\n",
    "In [problem 1](https://projecteuler.net/problem=1), we applied the following formula for [triangular numbers](https://en.wikipedia.org/wiki/Triangular_number):\n",
    "$$1 + 2 + 3 + \\cdots + n = \\frac{n(n+1)}{2}$$\n",
    "We can apply it again here and determine that\n",
    "$$(1 + 2 + 3 + \\cdots + 100)^2 = \\left(\\frac{100(101)}{2}\\right)^2 = 25502500$$\n",
    "\n",
    "A similar formula exists for computing [sums of squares](https://en.wikipedia.org/wiki/Square_pyramidal_number):\n",
    "$$1^2 + 2^2 + 3^2 + \\cdots + n^2 = \\frac{n(n+1)(2n+1)}{6}$$\n",
    "(In fact, [Faulhaber's formula](https://en.wikipedia.org/wiki/Faulhaber%27s_formula) gives a formula for the sum of $k$th powers, but we obviously only need the cases $k=1$ and $k=2$ for this problem.) Consequently,\n",
    "$$1^2 + 2^2 + 3^2 + \\cdots + 100^2 = \\frac{100(101)(201)}{6} = 338350$$\n",
    "\n",
    "Therefore, the difference is $25502500 - 338350 = 25164150$."
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