Question

Let $nϵN$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1,2n+2,...,$ up to $n^{th}$ number are $A_n$, $G_n$, $H_n$ and $R_n$ respectively.
On the basis of above information answer the following questions$\underset{n\to \infty }{lim}{\left(\frac{R_n}{n}\right)}^2$ equals

• A. $\frac{17}{3}$
• B. $\frac{13}{3}$
• C. $\frac{11}{3}$
• D. $\frac{19}{3}$

Answer 1

The correct option is D $\frac{19}{3}$

$R_n=$ root mean square of $2n+1,2n+2,...,2n+n$

$\Rightarrow R_n={\left\{\frac{{(2n+1)}^2+{(2n+2)}^2+...+{(2n+n)}^2}{n}\right\}}^{1/2}$

$\Rightarrow$ $R_n^2=\frac{1}{n}[{(2n+1)}^2+{(2n+2)}^2+...+{(2n+n)}^2]$

$\Rightarrow {\left(\frac{R_n}{n}\right)}^2=\frac{1}{n}\sum _{r=1}^n{(2+\frac{r}{n})}^2$
$\therefore$ $\underset{n\to \infty }{lim}\frac{R_n^2}{n^2}=\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{n}{(2+\frac{r}{n})}^2$

${\int }_0^1{(2+x)}^{2}dx=\frac{1}{3}{\left({(2+x)}^3\right)}_0^1=\frac{1}{3}(27-8)$

$\therefore$ $\underset{n\to \infty }{lim}{\left(\frac{R_n}{n}\right)}^2=\frac{19}{3}$