\[ \text { If }\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots \] prove that, \( a_{...
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0\[
\text { If }\left(1+x+x^{2}\right)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots
\]
prove that, \( a_{r}-n a_{r-1}+\frac{n(n-1)}{1.2} a_{r-2}-\ldots+(-1)^{p} \frac{n !}{r !(n-r) !} a_{0}=0 \), unless \( r \) is a multiple of 3 . What is its value in this case?
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