Let w be a complex number such that w^3=1 Find all possible values of 1/(1+w) + 1/(1+w^2).
+26367 Let \(w\) be a complex number such that \(w^3=1\) Find all possible values of \(\dfrac{1}{1+w} + \dfrac{1}{1+w^2}\).
\(w^3 = ww^2 ~\text{ or }~ w^2=\dfrac{w^3}{w} \)
\(\begin{array}{|rcll|} \hline && \mathbf{\dfrac{1}{1+w} + \dfrac{1}{1+w^2}} \quad & | \quad w^2=\dfrac{w^3}{w} \\\\ &=& \dfrac{1}{1+w} + \dfrac{1}{1+\dfrac{w^3}{w}} \\\\ &=& \dfrac{1}{1+w} + \dfrac{w}{w+w^3} \quad & | \quad w^3=1 \\\\ &=& \dfrac{1}{1+w} + \dfrac{w}{w+1} \\\\ &=& \dfrac{1+w}{1+w} \\\\ &=& \mathbf{1} \\ \hline \end{array}\)
