prove that `sum_(r=0)^n 3^r nC_r=4^n`

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

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

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

Prove that sum_(r=0)^(n)(-1)^(r)nC_(r)[(1)/(2^(r))+(3)/(2^(2r))+(7)/(2^(3r))+(15)/(2^(4r))+...up to mterms]=(2^(mn)-1)/(2^(mn)(2^(n)-1))

Prove that sum_(r=0)^(n)r(n-r)C_(r)^(2)=n^(2)(^(2n-2)C_(n))

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

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

Evaluate sum_(r = 0)^(n) 3^r ""^nC_r