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If n in Nsuch that is not a multiple of...

If` n in N`such that is not a multiple of 3 and`(1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r )`, then find the value of `sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r )`.

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To solve the problem, we need to evaluate the expression: \[ \sum_{r=0}^{n} (-1)^{r} a_{r} \binom{n}{r} \] where \( a_{r} \) are the coefficients from the expansion of \( (1 + x + x^2)^{n} \). ...
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