If `(1+x+x^(2))^(n)=1 +a_(1)x+a_(2)x^(2)+a_(3)x^(3)`
`+……..+a_(2n).x^(2n)` then prove that:
`(i) a_(1)+a_(3)+a_(5)+…..+a_(2n-1) =(3^(n)-1)/(2)`
`(ii) a_(2)+a_(4)+a_(6)+……+a_(2n)=(3^(n)-1)/(2)`

Explanation

To solve the given problem, we need to prove two statements regarding the coefficients \( a_1, a_2, \ldots, a_{2n} \) of the expansion of \( (1 + x + x^2)^n \).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression:
\[
(1 + x + x^2)^n = 1 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_{2n} x^{2n}
...