Prove that: `1+^3C_1+^4C_2=^5C_3`

Prove that: .^2C_1+^3C_1+^4C_1=^5C_3-1

Prove that 1+""^(3)C_(1)+ ""^(4)C_(2)= ""^(5)C_(3) .

Prove that 3C_(1)+7C_(2)+11C_(3)+.........+(4n-1)C_(n)=1+(2n-1)*2^(n)

Prove that (i) C_(1)+2C_(2)+3C_(3)+……+nC_(n)=n.2^(n+1) (ii) C_(0)+(C_(1)/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

If C_1, C_2 , C_3 , C_4 are the coefficients of any consecutive terms in the expansion of (1+x)^n , prove that : (C_1)/(C_1+C_2) + (C_3)/(C_3+C_4) = (2C_2)/(C_2 + C_3)

Prove that : ""^(2)C_(1)+ ""^(3)C_(1)+""^(4)C_(1)=""^(3)C_(2)+""^(4)C_(2) .

Prove that : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+...+2n.C_(n)=1+n*2^(n)

"if "(1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+…….+2n.C_(n)=1+n.2^(n)

If (1+x)^n=C_0+C_1x+C_2x^2+C_3x^3+...+C_nx^n then prove that 2.C_0+2^2C_1/2+2^3C_2/3+2^4C_3/4+...+2^(n+1)C_n/(n+1)=(3^(n+1)-1)/(n+1)