Question

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 $10\left(n^{2}-n\right)$ $( n \in N , n >1)$ are three consecutive terms of a $G.P$., then $n$ is equal to_____

Answer 1

$\int\limits_{0}^{n}\{x\} d x=n \int\limits_{0}^{1}\{x\} d x=n \int\limits_{0}^{1} x d x=\frac{n}{2}$
$\int\limits_{0}^{n}[x] d x=\int\limits_{0}^{n}(x-\{x\}) d x=\frac{n^{2}}{2}-\frac{n}{2}$
$\Rightarrow \left(\frac{n^{2}-n}{2}\right)^{2}=\frac{n}{2} \cdot 10 \cdot n(n-1)($ where $n>1)$
$\Rightarrow \frac{n-1}{4}=5 \Rightarrow n=21$