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If n is not a multiple of 3, then the coefficient of `x^n` in the expansion of `log_e (1+x+x^2)`is : (A) `1/n` (B) `2/n` (C) `-1/n` (D) `-2/n`
A. `(-2)/(n)`
B. `(2)/(n)`
C. `(1)/(n)`
D. none of these

1 Answer

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Best answer
Answer:
We have
`log(1+x+x^(2))=logP{(1-wx)(1-w^(2)x)}`
`rarr log(1+x+x^(2))=log(1-wx)+log(1-w^(2)x)`
`rarr log(1+x+x^(2))=log(1-wx)+log(1-w^(2)x)`
`rarr log(1+x+x^(2))=-underset(r=1)overset(infty)Sigma(w^(r )+w^(2r))(x^(r ))/(r )`
`therefore "coefficient of" x^(n) log(1+x+x^(2))` is equal to
`-(1)/(n)(w^(n)+w^(2n))=(2)/(n)`

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