Ceil The Limit!

First, we use sandwich theorem to simplify the sum to $\large \lim_{n\to\infty} \dfrac{ x + 2x + \cdots + n x }{n^2}$

This can be solved using sum of 1st n natural numbers and taking its limit which turns out to be $\frac{x}{2}$ So answer is 0.25

$\begin{aligned} L & = \lim_{n \to \infty} \frac {\lceil x \rceil +\lceil 2x \rceil + \lceil 3 x \rceil + \cdots + \lceil n x \rceil}{n^2} \\ & = \lim_{n \to \infty} \frac {x+2x+3x+\cdots + nx}{n^2} \\ & = \lim_{n \to \infty} \frac {\frac {n(n+1)x}2}{n^2} \\ & = \lim_{n \to \infty} \frac {(n+1)x}{2n} \\ & = \lim_{n \to \infty} \left(\frac x2 + \frac x{2n} \right) \\ & = \frac x2 \end{aligned}$

$\implies a^2 + b^2 = \left(\frac 12\right)^2 + 0^2 = \frac 14 = \boxed{0.25}$