Prove that `^n C_0+^n C_3+^n C_6+=1/3(2^n+2cos(npi)/3)` .

Prove that: ^(2n)C_0-3.^(2n)C_1+3^2.^(2n)C_2-..+(-1)^(2n) ..3^(2n)^(2n)C_(2n)=4^n for all value of N

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

Prove that: :2^(n)C_(n)=(2^(n)[1.3.5(2n-1)])/(n!)

If (1+x)^n = C_0 + C_1x + C_2x^2 + ………. + C_n x^n , prove that : C_0 + (C_1)/(2) + (C_2)/(3) + ……. + (C_n)/(n+1) = (2^(n+1) -1)/(n+1)