Sum of series

We simply rewrote this series to $f(x) = \begin{cases} x & \text{ if } \sqrt{x} \notin \mathbb{N} \\ -x & \text{ if } \sqrt{x} \in \mathbb{N} \end{cases}$ For $1 \leq x \leq 100$ , there will be 10 cases where $\sqrt{x} \in \mathbb{N}$ . We can then wrote the series to

$(1+2+3+4+\dots+100)-2(1+4+9+16+\dots+100)$

Which is equal to $\frac{100(101)}{2}-2(\frac{10(11)(21)}{6}) = 5050-2(385) = 4280$

Let the sum be $S$ . Then we have:

$\begin{aligned} S & = \color{#D61F06}{-1} + 2 + 3 +...+ 8 \color{#D61F06}{-9} + 10 +... + 15 \color{#D61F06}{-16} + 17+... + 24 \color{#D61F06}{-25} + 26 + ... + 99 \color{#D61F06}{-100} \\ & = (1+2+3+4+...+100) - 2\color{#D61F06}{(1^2 + 2^2 + 3^2 + ... + 10^2)} \\ & = \sum_{n=1}^{100} n - 2 \color{#D61F06}{\sum_{n=1}^{10} n^2} \\ & = \frac{100\times 101}{2} - 2 \color{#D61F06}{\left(\frac{10\times 11 \times 21}{6} \right)} \\ & = 5050 - 2 \color{#D61F06}{\left(385 \right)} \\ & = \boxed{4280} \end{aligned}$