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

Find the sum sum_(r=0)^n^(n+r)C_r .

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

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

Prove that .^(n)C_(0) - ^(n)C_(1) + .^(n)C_(2)- ^(n)C_(3) + "…" + (-1)^(r) + .^(n)C_(r) + "…" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that sum_(r=1)^k(-3)^(r-1)^(3n)C_(2r-1)=0,w h e r ek=3n//2 and n is an even integer.

Prove that sum_(r=0)^n(-1)^r^n C_r[1/(2^r)+3/(2^(2r))+7/(2^(3r))+(15)/(2^(4r))+ u ptomt e r m s]=(2^(m n)-1)/(2^(m n)(2^n-1))

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

Prove that ""^(n)C_r + ""^(n)C_(r-1) = ""^(n+1)C_r

Find the sum sum_(j=0)^(n) (""^(4n+1)C_(j)+""^(4n+1)C_(2n-j)) .

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),w h e r ei=sqrt(-1)dot