Prove that : Prove that
`C_(0)+3.C_(1)+3_(2).C_(2)+……+3^(n).C_(n)=4^(n)`

Prove the C_(0)+2.C_(1)+4.C_(2)+8.C_(3)+…+2^(n).C_(n)=3^(n)

Prove that : Prove that 2.C_(0)+7.C_(1)+12.C_(2)+……..+(5n+2)C_(n)=(5n+4)2^(n-1).

Prove that following C_(0)+2.C_(1)+4.C_(2)+8.C_(3)+…+2^(n).C_(n)=3^(n)

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

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that Prove that : C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+….+C_(n)^(2)=""^(2n)C_(n)

Prove that 3.C_0 + 6.C_1 + 12.C_2 + ………+3.2^n.C_n = 3^(n+1)

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

Prove that C_0 + 2.C_1 + 4.C_2 + 8.C_3 + ……+2^n.C_n = 3^n

With usual notations prove that C_0 + 3.C_1 + 3^2.C_2 + ………..+3^n .C_n = 4^n

Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).