Binomial Theorem

For $$ n\in N$$, we put $${ \left( 1+x+{ x }^{ 2 } \right)  }^{ n }=\sum _{ r=0 }^{ 2n }{ { a }_{ r }{ x }_{ r } } \quad .....(1)$$
If n is positive integer and if $$\displaystyle \left ( 1+4x+4x^{2} \right )^{n}=\sum_{r=0}^{2n}a_{r}x^{r}$$ where $$\displaystyle a_{i}'s$$ are (i  0, 1, 2, 3, ....., 2n) real numbers
Sum of coefficients in the expression of $$(x+2y+z)^{10}$$ is
The coefficient of the term independent of y in the expansion of $$\left [  \dfrac {y \, + \, 1} {y^{2/3} \, - \, y^{1/3}  \, + 1} \, - \, \dfrac{y \, - \, 1}{y \, - \, y^{1/3}} \right] ^{10} $$ is $$25$$. 
In the usual notations prove that 
$$C_1 \, + \, 2C_2 \, x \, + \, 3C_3 \, x^2 \, + \, ..... \, nC_n \, x^{n \, - \, 1} \, = \, n(1 \, + \, x){n \, - \, 1}$$