Let x and [x] denote the fractional part of x and the greatest integer x respectively of a…

Question

Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\le x$ respectively of a real number $x$. If $\displaystyle\int_0^n\{x\}\,dx$, $\displaystyle\int_0^n[x]\,dx$ and $10(n^2-n)$, ($n\in N,n>1$) are three consecutive terms of a G.P., then $n$ is equal to\_\_\_\_\_\_\_.

Answer: 21

Accepted Answer

$$\int_0^n\{x\}dx=n\int_0^1x\,dx=\frac n2,\qquad \int_0^n[x]dx=\int_0^n(x-\{x\})dx=\frac{n^2}2-\frac n2$$

Since these along with $10(n^2-n)$ are in G.P.:
$$\left(\frac{n^2-n}2\right)^2=\frac n2\cdot10(n^2-n)\Rightarrow n^2=21n\Rightarrow n=21$$

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