$\sum x _{0}^{1}=\frac{3\left(1-\left(\frac{1}{2}\right)\right)^{20}}{1 \frac{-1}{2}}=6\left(1-\frac{1}{2^{20}}\right)$

$=\sum_{ i =1}^{20}\left( x _{ i – i }\right)^{2}$

$=\sum_{ i =1}^{20}\left( x _{ i }\right)^{2}+( i )^{2}-2 x _{ i } i$

Now $=\sum_{i=1}^{20}\left(x_{i}\right)^{2}=\frac{9\left(1-\left(\frac{1}{4}\right)\right)^{20}}{1-\frac{1}{4}}=12\left(1-\frac{1}{2^{40}}\right)$

$\sum_{i=1}^{20} i^{2}=\frac{1}{6} \times 20 \times 21 \times 41=2870$

$\sum_{ i =1}^{20} x _{ i } i = s =3+2.3 \frac{1}{2}+3.3 \frac{1}{2^{2}}+4.3 \frac{1}{2^{3}}+\ldots \ldots AGP$

$=6\left(2-\frac{22}{2^{20}}\right)$

$\overline{ x }=\frac{12-\frac{12}{2^{40}}+2870-12\left(2-\frac{22}{2^{20}}\right)}{20}$

$\overline{ x }=\frac{2858}{20}+\left(\frac{-12}{2^{40}}+\frac{22}{2^{20}}\right) \times \frac{1}{20}$

${[\overline{ x }]=142 }$