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$\underset{n\to \infty }{lim}\frac{A_n}{n}$ equals

• A. $\frac{5}{2}$
• B. $\frac{2}{5}$
• C. $\frac{3}{5}$
• D. $\frac{5}{3}$

Answer 1

Answer: (A)

$\frac{5}{2}$

Given numbers are $2n+1,2n+2,...,2n+n$

$\therefore$ $A_n=$ A.M. of $2n+1,2n+2,...,2n+n$

$\Rightarrow A_n=\frac{1}{n}\{(2n+1)+(2n+2)+...+(2n+n)\}$

$\Rightarrow A_n=\frac{1}{n}[2n\left(n\right)+(1+2+3+...+n)]$

$\Rightarrow A_n=\frac{1}{n}\{2n^2+\frac{n(n+1)}{2}\}$
$\Rightarrow A_n=\frac{5n^2+n}{2n}$

$\therefore$ $\frac{A_n}{n}=\frac{5}{2}+\frac{1}{2n}$

$\therefore$ $\underset{n\to \infty }{lim}\frac{A_n}{n}=\frac{5}{2}$