#### Passage

Let $n \space\epsilon \space N$ & 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
Mathematics

# $\displaystyle \lim_{n\rightarrow \infty }\frac{A_{n}}{n}$ equals

$\displaystyle \int_{0}^{1}\left ( 2+x \right )dx$

##### SOLUTION
$A_{n}=$ A.M. of $\left ( 2n+1 \right ), \left ( 2n+2 \right ), ..., \left ( 2n+n \right )$
$\therefore$   $\displaystyle A_{n}=\dfrac{\left ( 2n+1 \right )+\left ( 2n+2 \right )+...+\left ( 2n+n \right )}{n}$
$\therefore$   $\displaystyle \dfrac{A_{n}}{n}=\dfrac{1}{n}\left [ \dfrac{\left ( 2n+1 \right )+\left ( 2n+2 \right )+...+\left ( 2n+n \right )}{n} \right ]$
$\Rightarrow A_{n} \displaystyle =\frac{1}{n}\left [ \left ( 2+\frac{1}{n} \right )+\left ( 2+\frac{2}{n} \right )+...+\left ( 2+\frac{n}{n} \right ) \right ]$
$\Rightarrow A_{n} \displaystyle =\frac{1}{n}\sum_{r=1}^{n}\left ( 2+\frac{r}{n} \right )$

$\therefore$   $\displaystyle \lim_{n\rightarrow \infty }\frac{A_{n}}{n}=\lim_{n\rightarrow \infty }\sum_{r=1}^{n}\frac{1}{n}\left ( 2+\frac{r}{n} \right )=\int_{0}^{1}\left ( 2+x \right )dx$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

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Match the integrals of f(x) if

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Q5 Single Correct Medium
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