Consider the expression $$(1 + x + x^2)^n = C_0 + C_1x + C_2x^2+\cdots + C_{2n-1}x^{2n-1} + C_{2n}x^{2n}$$ (where $n$ belongs to positive integers),then the value of
$C_0 + C_3 + C_6 + C_9 + C_{12}+\cdots = 3^n/3$
I know it can be directly obtained by using complex number ω, but I was interested in getting the result by remaining in the real domain.I tried expressing each term of the series into sums of binomial coefficients and tried canceling terms ,rearranging them,used simple properties to add term to get some kind of pattern but could not get anywhere. Sincere thanks.
Hint: Try using induction. Note that by substituting $x = 1$ we have $$3^n = C_0 + C_1 + C_2 + \ldots + C_{2n}.$$ and by arranging $$\begin{array}{rllllll} (1 + x + x^2)^n (1 + x + x^2) &= C_0 &+ C_1 x &+ C_2 x^2 &+ C_3 x^3 &+ \ldots &+ C_{2n} x^{2n} \\ &= & + C_0 x & + C_1 x^2 & + C_2 x^3 &+ C_3 x^4 &+ \ldots & + C_{2n} x^{2n+1} \\ &= & & + C_0 x^2 &+ C_1 x^3 & + C_2 x^4 &+ C_3 x^5 &+ \ldots & + C_{2n} x^{2n + 2}. \end{array}$$