If `(1 + x)^(n) = C_(0) = C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)` , find the values of the following `(sumsum)_(0leilt j le n)jC_(i)`

Explanation

` underset(0leiltjlen)(sumsum) j C_(i) = sum_(r=0)^(n-1)""^(n)C_(r)`
` {( r + 1)+ (r + 2) + (r+3)+ ...+ n}`
`sum_(r=0)^(n-1)""^(n)C_(r)*((n-r)(n+r +1))/(2)`
` = 1/2sum_(r=0)^(n-1)""^(n)C_(r)(n^(2) - r^(2) + n-r)`
`=1/2(n^(2) + n) sum_(r=0)^(n-1) ""^(n)C_(r) - 1/2sum_(r=0)^(n-1)r.""^(n)C_(r) - 1/2sum_(r=0)^(n-1) r^(2) .""^(n)C_(r)`
`=1/2(n^(2) + n) (2^(n) -1) -1/2*n*n(2^(n-1)-1)- 1/2*n[(n-1)(n^(n-2)-1) + 2^(n-2)-1]`
` = (3n+1)*2^(n-3)` .