#6 - Sum square difference
The sum of the squares of the first ten natural numbers is, \[1^2 + 2^2 + \cdots + 10^2 = 385\]
The square of the sum of the first ten natural numbers is, \[(1 + 2 + \cdots + 10)^2 = 55^2 = 3025\]
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.
Just directly loop until 100. No special packages are necassary, and we can use list comprehension to make the code extra short.
# file: "problem006.py"
n = 100
sum_square = sum(x ** 2 for x in range(n + 1))
square_sum = sum(range(n + 1)) ** 2
print(square_sum - sum_square)
Running gives,
25164150
5.570007488131523e-05 seconds.
Bonus
We can also do this analytically. The expression we are asked to solve is \[S = \left( \sum_{i=1}^{100} i \right)^2 - \sum_{i=1}^{100} i^2\]
Recall that $\sum_{i=1}^n i = \frac{n(n+1)}{2}$ and $\sum{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$. Substituting and solving, we get \[\begin{aligned} S &= \left( \frac{100(100+1)}{2} \right)^2 - \frac{100(100 + 1)(2(100) + 1)}{6} \\ &= 5050^2 - 338350 \\ &= \boxed{25164150} \end{aligned}\]
