Let {x} and [x] denote the fractional part


Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real number $x$. If $\int_{0}^{n}\{x\} d x, \int_{0}^{n}[x] d x$ and $10\left(n^{2}-n\right),(n \in \mathbf{N}, n>1)$ are three consecutive terms of a G.P., then $n$ is equal to______.


$\int_{0}^{n}\{x\} d x=n \int_{0}^{1} x \cdot d x=\frac{n}{2}$

$\Rightarrow \int_{0}^{n}[x] d x=\int_{0}^{n}(x-\{x\}) d x=\frac{n^{2}}{2}-\frac{n}{2}$

According to the questions,

$\frac{n}{2}, \frac{n^{2}-n}{2}, 10\left(n^{2}-n\right)$ are in GP

$\therefore\left(\frac{n^{2}-n}{2}\right)^{2}=\frac{n}{2} \times 10\left(n^{2}-n\right)$

$\Rightarrow n^{2}=21 n \Rightarrow n=21$

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