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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


ANSWER

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


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
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