Given,

Mean of $x_1, x_2..........,x_{100}$ is $200$

So, the sum of observation will be,

$\sum x_i=100\times 200$

Now using the sum of $A.P$ formula in above equation as all terms are in arithmetic progression we get,

$\frac{100}{2}\left(x_1+x_{100}\right)=100\times 200$

$\Rightarrow 50\left(2+x_{100}\right)=100\times 200$

$\Rightarrow x_{100}=398$

$\Rightarrow x_1+99d=398$

$\Rightarrow d=4$

Now, $x_i=2+(i-1)4=4i-2$

So, $y_i=i\left(x_i-i\right)=3i^2-2i$

Now finding mean we get,

$\overline{y}=\frac{1}{100}\sum y_i$

$\Rightarrow \overline{y}=\frac{1}{100}\sum \left(3i^2-2i\right)$

$\Rightarrow \overline{y}=\frac{1}{100}\left[3\times \frac{100\times 101\times 201}{6}-2\frac{100\times 101}{2}\right]$

$\Rightarrow \overline{y}=\left[\frac{101\times 201}{2}-101\right]$

$\Rightarrow \overline{y}=10049.5$