Let x _ 1 , x _ 2 , , x _ 100 be in an arithmetic progression, with...

Given,

Mean of ${x}_{1}, {x}_{2} . . . . . . . . . . , {x}_{100}$ is $200$

So, the sum of observation will be,

$∑ {x}_{i} = 100 \times 200$

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

$\dfrac{100}{2}\left({x}_{1} + {x}_{100}\right) = 100 \times 200$

$⇒ 50\left(2 + {x}_{100}\right) = 100 \times 200$

$⇒ {x}_{100} = 398$

$⇒ {x}_{1} + 99d = 398$

$⇒ d = 4$

Now, ${x}_{i} = 2 + ( i - 1 ) 4 = 4i - 2$

So, ${y}_{i} = i\left({x}_{i} - i\right) = 3{i}^{2} - 2i$

Now finding mean we get,

$\overset{¯}{y} = \dfrac{1}{100} ∑ {y}_{i}$

$⇒ \overset{¯}{y} = \dfrac{1}{100} ∑ \left(3{i}^{2} - 2i\right)$

$⇒ \overset{¯}{y} = \dfrac{1}{100}\left[3 \times \dfrac{100 \times 101 \times 201}{6} - 2\dfrac{100 \times 101}{2}\right]$

$⇒ \overset{¯}{y} = \left[\dfrac{101 \times 201}{2} - 101\right]$

$⇒ \overset{¯}{y} = 10049 . 5$