The value of $\mathop {\lim }\limits_{n \to \infty } {{[r] + [2r] + ... + [nr]} \over {{n^2}}}$, where r is a non-zero real number and [r] denotes the greatest integer less than or equal to r, is equal to :

We know,<br><br>(x $-$ 1) $\le$ [x] &lt; x<br><br>$\therefore$ (r $-$ 1) $\le$ [r] &lt; r <br><br>(2r $-$ 1) $\le$ [2r] &lt; 2r <br><br>.<br><br>.<br><br>.<br><br>(nr $-$ 1) $\le$ [nr] &lt; nr<br><br>Adding<br><br>$${{n(n + 1)} \over 2}r - n \le [r] + [2r] + .......[nr] &lt; {{n(n + 1)} \over 2}r$$ <br><br>$${{{{n\left( {n + 1} \right)} \over 2}r - n} \over {{n^2}}} \le {{\left[ r \right] + \left[ {2r} \right] + .... + \left[ {nr} \right]} \over {{n^2}}} \le {{{{n\left( {n + 1} \right)} \over 2}r} \over {{n^2}}}$$ <br><br>$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{{n(n + 1)} \over 2}r - n} \over {{n^2}}}} \right) \le L &lt; \mathop {\lim }\limits_{n \to \infty } {{n(n + 1)} \over 2}r$$<br><br>$\Rightarrow {r \over 2} \le L &lt; {r \over 2}$<br><br>$\Rightarrow L = {r \over 2}$