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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