Let {x} and [x] denote the fractional part of x and
the greatest integer $\le$ x respectively of a real
number x. If $\int_0^n {\left\{ x \right\}dx} ,\int_0^n {\left[ x \right]dx}$ and 10(n² – n),
$\left( {n \in N,n > 1} \right)$ are three consecutive terms of a G.P., then n is equal to_____.

$$\int\limits_0^n {\left\{ x \right\}} dx = n\int\limits_0^1 x dx = n\left( {{{{x^2}} \over 2}} \right)_0^1 = {n \over 2}$$ <br><br>[As period of {x} = 1] <br><br>$$\int\limits_0^n {\left[ x \right]} dx = \int\limits_0^1 0 dx + \int\limits_1^2 1 dx + ... + \int\limits_{n - 1}^n {\left( {n - 1} \right)} dx$$ <br><br>= 1 + 2 + 3 + ....+ (n - 1) <br><br>= ${{n\left( {n - 1} \right)} \over 2}$ <br><br>As ${n \over 2}$, ${{n\left( {n - 1} \right)} \over 2}$, 10(n² – n) are in GP. <br><br>$\therefore$ $${\left[ {{{n\left( {n - 1} \right)} \over 2}} \right]^2} = {n \over 2} \times 10\left( {{n^2} - n} \right)$$ <br><br>$\Rightarrow$ n² = 21n <br><br>$\Rightarrow$ n = 21