Prove that `sum_(r=0)^n3^r^n C_r=4^n`.

Prove that sum_(r=0)^n^n C_r3^r=4^n

Prove that sum_(r = 0)^n r^3 . C_r = n^2 (n +3).2^(n-3)

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot

Prove that sum_(r=0)^n^n C_r(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4), where i=sqrt(-1)dot

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

Prove that sum_(r=0)^(n) 3^( r" "n)C_(r ) =4^(n) .

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

If x+y=1, prove that sum_(r=0)^n .^nC_r x^r y^(n-r) = 1 .

Prove that sum_(r=0)^(n) ""^(n)C_(r).(n-r)cos((2rpi)/(n)) = - n.2^(n-1).cos^(n)'(pi)/(n) .