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  4. Let Xn denote the mean of first n natural numbers, then the mean of X1,X2, ldots ,X100 is

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Q. Let $X_{n}$ denote the mean of first $n$ natural numbers, then the mean of $X_{1},X_{2},\ldots ,X_{100}$ is

NTA AbhyasNTA Abhyas 2022

Solution:

$X_{n}=\frac{ \displaystyle\sum _{r = 1}^{n} r}{n}=\frac{n \left(n + 1\right)}{2 n}=\frac{n + 1}{2}$
$\frac{ \displaystyle\sum _{n = 1}^{100} X_{n}}{n}=\frac{\frac{100 \times 101}{2} + 100}{200}$
$=\frac{101}{4}+\frac{1}{2}=25.75$