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

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

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)

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

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

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

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)

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

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

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