Prove that `.^(n)C_(1) + 2 .^(n)C_(2) + 3 .^(n)C_(3) + "…." + n . ^(n)C_(n) = n 2^(n-1)`.

Explanation

Method I :
`.^(n)C_(1) + 2.^(n)C_(2)+3.^(n)C_(3)+"….."+n.^(n)C_(n)`
`=underset(r=1)overset(n)sumr.^(n)C_(r)`
`= n underset(r=1)overset(n)sum.^(n-1)C_(r-1)`
`= n(.^(n-1)C_(0) + .^(n-1)C_(1) + .^(n-1)C_(2) + .^(n-1)C_(3)+"…."+.^(n-1)C_(n-1))`
`= n2^(n-1)`
Method II :
We have `(1+x)^(n) = .^(n)C_(0) + .^(n)C_(1)x+.^(n)C_(2)x^(2)+"...."+.^(n)C_(n)x^(n)`
Differentiating w.r.t x, we get
`n(1+x)^(n-1)= .^(n)C_(1)+2 xx .^(n)C_(2)x + 3 xx .^(n)C_(3)x^(2) +"......" + n xx .^(n)C_(n)x^(n-1)`
Putting `x = 1`, we get
`n2^( n-1) = .^(n)C_(1)+2xx .^(n)C_(2) + "....." + n xx .^(n)C_(n)`
Using `A.M. ge G.M.`, we get
`(.^(n)C_(1)+2.^(n)C_(2)+3.^(n)C_(3)+".....+n.^(n)C_(n))/((n(n+1))/(2))`
`[(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)]^((1)/((pi(n-1))/2))`
`rArr (n2^(n-1))/((n(n+1))/(2))ge [(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)]^((2)/(n(n+1)))`
`(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)le((2^(n))/(n+1))^((n(n+1))/(2))`
or `(.^(n)C_(1))(.^(n)C_(2))^(2)(.^(n)C_(3))^(3)"....."(.^(n)C_(n))^(n)le((2^(n))/(n+1))^(.^(n+1)C_(2))`