<s>[INST] Find the sum of the even-degree coefficients I am doing practice problems from the AMC-10 math competition. I do not understand how to go about solving the problem
Let $(1+x+x^2)^n = a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}$ be an identity of $x$. If we let $s = a_0 + a_2 + a_4 + ... + a_{2n}$, find $s$.
The choices are $A, 2^n$, $B, (2^n) + 1$, $C, \frac{3^n - 1}{2}$ , $D, \frac{3^n}{2}$ , and $E, \frac{3^n + 1}{2}$
Thanks in advance
 [/INST] As noted by lab in a comment, for $x=1,-1$ you get
$$3^n = a_o+a_1+\ldots+a_{2n}  \\ 1 = a_o-a_1+\ldots+a_{2n}$$
so $3^n+1 = 2a_0+ 2a_2+\ldots+2a_{2n}$. 
Another (lazy) idea if you already have some candidates: for $n=1, a_0+a_2=2$, for $n=2, (1+x+x^2)^2 = 1+2x+3x^2+2x^3+x^4$, so $a_0+a_2+a_4 = 5$. Then the only candidate possible is $\frac{3^n+1}{2}$.
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