Find the sum `sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j)`

Explanation

`S=underset(inej)(sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j))=sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j)-underset(i=j)(sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j))`
`=(sum_(i=1)^(10)i)(sum_(j=1)^(10)2^(j))-sum_(i=1)^(10)ixx2^(i)`
`=((10xx11)/2)((2(2^(10)-1))/(2-1))-sum_(i=1)^(10)xx2^(i)`
Let `S_(1)=sum_(i=1)^(10)ixx2^(i)`
`thereforeS_(1)=1xx2+2xx2^(2)+3xx2^(3)+...+10xx2^(10)`
`therefore 2s_(1)=1xx2^(2)+2xx2^(3)+3xx2^(4)+...+10xx2^(11)`
Subtracting, we get
`-S_(1)=(2+2^(2)+2^(3)+...+2^(10))-10xx2^(11)`
`therefore S_(1)=10xx2^(11)-2(2^(10)-1)=9xx2^(11)+2`
`thereforeS=110xx(2^(10)-1)-(9xx2^(11)+2)`
`=5xx2^(12)-108`