Question

please tell the answer with explanation
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Answer 1

Answer:

Step-by-step explanation:

We have,

$\sf{\bar{x}\,\,is\,\,the\,\,mean\,\,of\,\,{x}_{1},\,{x}_{2},\,{x}_{3},\,\cdots,\,{x}_{n}}$

So,

$\rm{\bar{x}=\dfrac{x_{1}+x_{2}+x_{3}+\cdots+x_{n}}{n}}$

$\rm{\implies\,n\bar{x}=x_{1}+x_{2}+x_{3}+\cdots+x_{n}\,\,\,\,\,\,\,\,\,\,\,...(1)}$

Now, required mean,

$\rm{\bar{x}^{\prime}=\dfrac{a\,x_{1}+a\,x_{2}+a\,x_{3}+\cdots+a\,x_{n}+\dfrac{x_{1}}{a}+\dfrac{x_{2}}{a}+\dfrac{x_{3}}{a}+\cdots+\dfrac{x_{n}}{a}}{2n}}$

$\rm{\implies\bar{x}^{\prime}=\dfrac{a\left(x_{1}+x_{2}+x_{3}+\cdots+x_{n}\right)+\dfrac{1}{a}\left(x_{1}+x_{2}+x_{3}+\cdots+x_{n}\right)}{2n}}$

From (1), we get,

$\rm{\implies\bar{x}^{\prime}=\dfrac{a\cdot\,n\,\bar{x}+\dfrac{1}{a}\cdot\,n\,\bar{x}}{2n}}$

$\rm{\implies\bar{x}^{\prime}=\dfrac{\left(a+\dfrac{1}{a}\right)\cdot\,n\,\bar{x}}{2n}}$

$\rm{\implies\bar{x}^{\prime}=\dfrac{\left(a+\dfrac{1}{a}\right)\cdot\,\bar{x}}{2}}$

$\rm{\implies\bar{x}^{\prime}=\left(a+\dfrac{1}{a}\right)\dfrac{\bar{x}}{2}}$