Let x_{1}, x_{2}, \ldots, x_{100} be in an arithmetic progression, with x_{1}=2 and their mean equal to 200 . If y_{i}=i\left(x_{i}-i\right), 1 \leq i \leq 100, then the mean of y_{1}, y_{2}, \ldots, y_{100} is :

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Answer: B. 10049.50 We have, mean of x_1, x_2 \ldots \ldots x_{100}=200 Where, x_1, x_2 \ldots x_{100} are in AP with first term as 2. \begin{aligned} 	ext { Mean } & =200 \\ & =\frac{\sum\limits_{i=1}^{100} x_i}{100}=200 \end{aligned} \begin{aligned} \frac{100}{2} 	imes[2 	imes 2+99 d] =20000 \\ \Rightarrow 4+99 d =400 \\ \Rightarrow 99 d =396 \\ d =4 \end{aligned} Also, \begin{aligned} y_i & =i\left(x_i-iight) \\ & =i[2+(i-1) 4-i] \\ & =i[3 i-2] \\ & =3 i^2-2 i \end{aligned} \begin{aligned} 	ext { Required mean } & =\frac{\sum\limits_{i=1}^{100} y_i}{100} \\ & =\frac{1}{100}\left[\sum_{i=1}^{100}\left(3 i^2-2 iight)ight] \\ & =\frac{1}{100}\left[\frac{3 	imes 100 	imes 101 	imes 201}{6}-2 	imes \frac{100 	imes 101}{2}ight]\\ & =\frac{20301}{2}-101 \\ & =10049.50 \end{aligned}

Let $$x_{1}, x_{2}, \ldots, x_{100}$$ be in an arithmetic progression, with $$x_{1}=2$$ and their mean equal to 200 . If $$y_{i}=i\left(x_{i}-i\right), 1 \leq i \leq 100$$, then the mean of $$y_{1}, y_{2}, \ldots, y_{100}$$ is :

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