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Statement1: if n in Na n dn is not a mu...

Statement1: if `n in Na n dn` is not a multiple of 3 and `(1+x+x^2)=sum_(r=0)^(2n)a_r x^r ,` then the value of `sum_(r=0)^n(-1)^r a r^n C_r` is zero Statement 2: The coefficient of `x^n` in the expansion of `(1-x^3)` is zero, if `n=3k+1orn=3k+2.`

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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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