Sum of some floored and summed series

Algebra Level 5

Let a series be defined such that

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

Let another series be defined as Tr=tr+r2.T_{r}= \left \lfloor t_{r}+r^{2} \right \rfloor .

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

n1n+(n+1)(2n+1)56n(n+1)(2n+1),n \left \lfloor \frac{1}{n}+(n+1)(2n+1) \right \rfloor - \frac{5}{6}n(n+1)(2n+1) ,

find the value of x\Large \color{#D61F06}{x}.

Given that n=100n=100

Details and Assumptions

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