Question

Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\le x$ respectively of a real number $x.$ If ${\int }_0^n\{x\}dx,{\int }_0^n[x]dx$ 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

Answer: 21

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