$\sqrt{\left( \displaystyle \sum_{i=1}^{100} (101-i)x_{i}\right) + 1 } + \sqrt{\left( \displaystyle \sum_{j=1}^{100} jx_{j} \right) + 2 } + \sqrt{\left( \displaystyle \sum_{k=1}^{100} 101x_{k}\right ) + 3 }$

If $x_{1}, x_{2}, \ldots, x_{100}$ are 100 positive reals such that $x_{1} + x_{2} + x_{3} + \cdots + x_{100} = 1$ , find the maximum value of the expression above.

Let it be of the form $a\sqrt{b}$ , where $b$ is square free and $a,b \in \mathbb{N}$ . Enter $(a+b)$ as your answer.

The answer is 43.

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