# Sum of some floored and summed series

Algebra Level 5

Let a series be defined such that

$t_{r+1}= \displaystyle \sum_{m=0}^r \left\lfloor \frac{(x+m)}{r+1} \right\rfloor .$

Let another series be defined as $T_{r}= \left \lfloor t_{r}+r^{2} \right \rfloor .$

if $$\displaystyle \sum_{r=1}^n T_{r}$$ can be written as

$n \left \lfloor \frac{1}{n}+(n+1)(2n+1) \right \rfloor - \frac{5}{6}n(n+1)(2n+1) ,$

find the value of $$\Large \color{Red}{x}$$.

Given that $$n=100$$

Details and Assumptions

• among the infinite values of $$x$$, the value asked for is also defined as one hundredth of the number of integral values of $$x$$
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