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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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`underset(r=0)overset(n)sum(-1)^(r).a_(r )..^(n)C_(r )`
`= a_(0) xx .^(n)C_(0) - a_(1) xx .^(n)C_(1) + a_(2) xx .^(n)C_(2)-"….."+(-1)^(n)a_(n) xx .^(n)C_(0)`
`=` Coefficient of `x^(n)` in `(1-x+x^(2))^(n)(x-1)^(n)`
`=` Coefficient of `x^(n)` in `(x^(3)-1)^(n)`
`= 0` (as n is a multiple of 3)
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